Research Article Optimal Lehmer Mean Bounds for the...
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Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 852516, 7 pages http://dx.doi.org/10.1155/2013/852516 Research Article Optimal Lehmer Mean Bounds for the Combinations of Identric and Logarithmic Means Xu-Hui Shen, 1 Wei-Ming Gong, 2 and Yu-Ming Chu 2 1 College of Nursing, Huzhou Teachers College, Huzhou 313000, China 2 School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China Correspondence should be addressed to Yu-Ming Chu; [email protected] Received 11 July 2013; Accepted 5 August 2013 Academic Editors: M. Coppens, Y. Miao, and P.-y. Nie Copyright © 2013 Xu-Hui Shen et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For any ∈ (0, 1), we answer the questions: what are the greatest values and and the least values and , such that the inequalities (, ) < (, ) 1− (,) < (, ) and (, ) < (, ) + (1 − )(, ) < (, ) hold for all , > 0 with ̸ =? Here, (, ), (, ), and (, ) denote the identric, logarithmic, and th Lehmer means of two positive numbers and , respectively. 1. Introduction For , > 0 with ̸ =, the logarithmic mean (, ) and identric mean (, ) are defined by (, ) = − log − log , (1) (, ) = 1 ( ) 1/(−) , (2) respectively. In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [1–19]. In [14, 17, 20], inequalities between , , and the classical arithmetic-geometric mean of Gauss are proved. e ratio of identric means leads to the weighted geometric mean ( 2 , 2 ) (, ) = ( ) 1/(+) , (3) which has been investigated in [11, 13, 21]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [22–24]. In [22], the authors study a variant of Jensen’s functional equation involving , which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in [5]. In [25, 26], it is shown that can be expressed in terms of Gauss’ hypergeometric function 2 1 . And in [26], the authors prove that the reciprocal of the logarithmic mean is strictly totally positive; that is, every × determinant with elements 1/( , ), where 0< 1 < 2 <⋅⋅⋅< and 0< 1 < 2 <⋅⋅⋅< , is positive for all ≥1. For ∈ R, the power mean (, ) and Lehmer mean (, ) of order of two positive numbers and with ̸ = are defined by (, ) = ( + 2 ) 1/ , (4) (, ) = +1 + +1 + . (5) It is well known that (, ) and (, ) are strictly increasing with respect to ∈ R for fixed , > 0 with ̸ =. e main properties for and are given in [27–32]. Let (, ) = ( + )/2, (, ) = √ , and (, ) = 2/(+) be the arithmetic, geometric, and harmonic means
Transcript of Research Article Optimal Lehmer Mean Bounds for the...
Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2013 Article ID 852516 7 pageshttpdxdoiorg1011552013852516
Research ArticleOptimal Lehmer Mean Bounds for the Combinations ofIdentric and Logarithmic Means
Xu-Hui Shen1 Wei-Ming Gong2 and Yu-Ming Chu2
1 College of Nursing Huzhou Teachers College Huzhou 313000 China2 School of Mathematics and Computation Science Hunan City University Yiyang 413000 China
Correspondence should be addressed to Yu-Ming Chu chuyuming2005126com
Received 11 July 2013 Accepted 5 August 2013
Academic Editors M Coppens Y Miao and P-y Nie
Copyright copy 2013 Xu-Hui Shen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
For any120572 isin (0 1) we answer the questions what are the greatest values119901 and120582 and the least values 119902 and120583 such that the inequalities119871119901(119886 119887) lt 119868
120572(119886 119887)119871
1minus120572(119886 119887) lt 119871119902(119886 119887) and 119871120582(119886 119887) lt 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 119871120583(119886 119887) hold for all 119886 119887 gt 0 with 119886 = 119887 Here
119868(119886 119887) 119871(119886 119887) and 119871119901(119886 119887) denote the identric logarithmic and 119901th Lehmer means of two positive numbers 119886 and 119887 respectively
1 Introduction
For 119886 119887 gt 0 with 119886 = 119887 the logarithmic mean 119871(119886 119887) andidentric mean 119868(119886 119887) are defined by
119871 (119886 119887) =
119886 minus 119887
log 119886 minus log 119887
(1)
119868 (119886 119887) =
1
119890
(
119887119887
119886119886)
1(119887minus119886)
(2)
respectively In the recent past both mean values havebeen the subject of intensive research In particular manyremarkable inequalities for 119871 and 119868 can be found in theliterature [1ndash19] In [14 17 20] inequalities between 119871 119868 andthe classical arithmetic-geometric mean of Gauss are provedThe ratio of identric means leads to the weighted geometricmean
119868 (1198862 1198872)
119868 (119886 119887)
= (119886119886119887119887)
1(119886+119887)
(3)
which has been investigated in [11 13 21] It might besurprising that the logarithmic mean has applications inphysics economics and even in meteorology [22ndash24] In[22] the authors study a variant of Jensenrsquos functionalequation involving 119871 which appears in a heat conduction
problem A representation of 119871 as an infinite product andan iterative algorithm for computing the logarithmic meanas the common limit of two sequences of special geometricand arithmetic means are given in [5] In [25 26] it is shownthat 119871 can be expressed in terms of Gaussrsquo hypergeometricfunction
21198651 And in [26] the authors prove that the
reciprocal of the logarithmic mean is strictly totally positivethat is every 119899 times 119899 determinant with elements 1119871(119886119894 119887119895)where 0 lt 1198861 lt 1198862 lt sdot sdot sdot lt 119886119899 and 0 lt 1198871 lt 1198872 lt sdot sdot sdot lt 119887119899 ispositive for all 119899 ge 1
For 119901 isin R the power mean 119872119901(119886 119887) and Lehmer mean119871119901(119886 119887) of order 119901 of two positive numbers 119886 and 119887with 119886 = 119887
are defined by
119872119901 (119886 119887) = (
119886119901+ 119887119901
2
)
1119901
(4)
119871119901 (119886 119887) =
119886119901+1
+ 119887119901+1
119886119901+ 119887119901
(5)
It is well known that 119872119901(119886 119887) and 119871119901(119886 119887) are strictlyincreasing with respect to 119901 isin R for fixed 119886 119887 gt 0 with 119886 = 119887The main properties for 119872119901 and 119871119901 are given in [27ndash32]
Let 119860(119886 119887) = (119886 + 119887)2 119866(119886 119887) = radic119886119887 and 119867(119886 119887) =
2119886119887(119886+119887) be the arithmetic geometric and harmonicmeans
2 Chinese Journal of Mathematics
of two positive numbers 119886 and 119887 respectively Then it is wellknown that
min 119886 119887 lt 119867 (119886 119887) = 119871minus1 (119886 119887)
= 119872minus1 (119886 119887) lt 119866 (119886 119887) = 119871minus12 (119886 119887)
= 1198720 (119886 119887) lt 119871 (119886 119887) lt 119868 (119886 119887) lt 119860 (119886 119887)
= 1198710 (119886 119887) = 1198721 (119886 119887) lt max 119886 119887
(6)
for all 119886 119887 gt 0 with 119886 = 119887The following sharp bounds for 119871 119868 (119871119868)12 and (119871+119868)2
in terms of power means are proved in [2ndash4 6 8 9 33]
11987223 (119886 119887) lt 119868 (119886 119887) lt 119872log 2 (119886 119887) (7)
1198720 (119886 119887) lt 119871 (119886 119887) lt 11987213 (119886 119887)
1198720 (119886 119887) lt radic119868 (119886 119887) 119871 (119886 119887) lt 11987212 (119886 119887)
1
2
(119868 (119886 119887) + 119871 (119886 119887)) lt 11987212 (119886 119887)
(8)
for all 119886 119887 gt 0 with 119886 = 119887Alzer and Qiu [19] proved that
120572119860 (119886 119887) + (1 minus 120572)119866 (119886 119887)
lt 119868 (119886 119887) lt 120573119860 (119886 119887) + (1 minus 120573)119866 (119886 119887)
119872119901 (119886 119887) lt
1
2
(119868 (119886 119887) + 119871 (119886 119887))
(9)
for all 119886 119887 gt 0 with 119886 = 119887 if and only if 120572 le 23 120573 ge 2119890 =
073575 and 119901 le log 2(1 + log 2) = 040938 The following sharp upper and lower Lehmer mean
bounds for 119871 119868 (119871119868)12 and (119871 + 119868)2 are presented in [34]
119871minus13 (119886 119887) lt 119871 (119886 119887) lt 1198710 (119886 119887) (10)
119872minus16 (119886 119887) lt 119868 (119886 119887) lt 1198710 (119886 119887) (11)
119871minus14 (119886 119887) lt 11986812
(119886 119887) 11987112
(119886 119887) lt 1198710 (119886 119887)
119871minus14 (119886 119887) lt
1
2
(119868 (119886 119887) + 119871 (119886 119887)) lt 1198710 (119886 119887)
(12)
for all 119886 119887 gt 0 with 119886 = 119887The purpose of this paper is to present the best possible
upper and lower Lehmer mean bounds of the product119868120572(119886 119887)119871
1minus120572(119886 119887) and the sum 120572119868(119886 119887)+ (1minus120572)119871(119886 119887) for any
120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
2 Lemmas
In order to establish ourmain result we need several lemmaswhich we present in this section
Lemma 1 If 119901 ge minus13 then the following statements are true
(1) 181199012+ 21119901 + 8 gt 0
(2) 2701199013+ 567119901
2+ 384119901 + 104 gt 0
(3) 11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448 gt 0
(4) 14581199015+4941119901
4+6903119901
3+4809119901
2+1714119901+320 gt 0
(5) 53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3+ 9672119901
2+
2264119901 + 320 gt 0
(6) 91199013+ 33119901
2+ 51119901 + 14 gt 0
Proof (1) We clearly see that
181199012+ 21119901 + 8 gt 18119901
2+ (minus
1
3
) times 21 + 8
= 181199012+ 1 gt 0
(13)
(2) Let
1198921 (119901) = 2701199013+ 567119901
2+ 384119901 + 104 (14)
Then simple computations lead to
1198921 (minus1
3
) = 29 gt 0
1198921015840
1(119901) = 6 (135119901
2+ 189119901 + 64)
gt 6 [1351199012+ 189 times (minus
1
3
) + 64]
= 6 (1351199012+ 1) gt 0
(15)
Therefore Lemma 1(2) follows from (14) and (15)(3) Let
1198922 (119901) = 11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448 (16)
Then simple computations yield
1198922 (minus1
3
) = 116 gt 0
1198921015840
2(119901) = 4536119901
3+ 10287119901
2+ 7758119901 + 1950
1198921015840
2(minus
1
3
) = 339 gt 0
11989210158401015840
2(119901) = 13608119901
2+ 20574119901 + 7758
gt 136081199012+ 20574 times (minus
1
3
) + 7758
= 136081199012+ 900 gt 0
(17)
Therefore Lemma 1(3) follows from (16) and (17)(4) Let
1198923 (119901) = 14581199015+ 4941119901
4+ 6903119901
3
+ 48091199012+ 1714119901 + 320
(18)
Chinese Journal of Mathematics 3
Then simple computations lead to
1198923 (minus1
3
) =
247
3
gt 0
1198921015840
3(119901) = 7290119901
4+ 19764119901
3+ 20709119901
2
+ 9618119901 + 1714
1198921015840
3(minus
1
3
) = 167 gt 0
11989210158401015840
3(119901) = 29160119901
3+ 59292119901
2+ 41418119901 + 9618
11989210158401015840
3(minus
1
3
) = 1320 gt 0
119892101584010158401015840
3(119901) = 87480119901
2+ 118584119901 + 41418
gt 874801199012+ 41418 + 118584 times (minus
1
3
)
= 874801199012+ 1890 gt 0
(19)
Therefore Lemma 1(4) follows from (18) and (19)(5) Let
1198924 (119901) = 53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+ 96721199012+ 2264119901 + 320
(20)
Then simple computations yield
1198924 (minus1
3
) =
284
3
gt 0 (21)
1198921015840
4(119901) = 32076119901
5+ 91125119901
4+ 109188119901
3
+ 659881199012+ 19344119901 + 2264
(22)
1198921015840
4(minus
1
3
) = 97 gt 0 (23)
11989210158401015840
4(119901) = 160380119901
4+ 364500119901
3
+ 3275641199012+ 131976119901 + 19344
(24)
11989210158401015840
4(minus
1
3
) = 228 gt 0 (25)
119892101584010158401015840
4(119901) = 108 (5940119901
3+ 10125119901
2+ 6066119901 + 1222) (26)
119892101584010158401015840
4(minus
1
3
) = 11340 gt 0 (27)
119892(4)
4(119901) = 1944 (990119901
2+ 1125119901 + 337) (28)
It follows from (28) and the discriminant Δ = (1125)2minus
4 times 990 times 337 = minus68895 lt 0 of the quadratic function 119891(119901) =
9901199012+ 1125119901 + 337 that
119892(4)
4(119901) gt 0 (29)
Therefore Lemma 1(5) follows from (20)ndash(27) and (29)
(6) Let
1198925 (119901) = 91199013+ 33119901
2+ 51119901 + 14 (30)
Then we have
1198925 (minus1
3
) =
1
3
gt 0
1198921015840
5(119901) = 27119901
2+ 66119901 + 51
gt 271199012minus 66 times (
1
3
) + 51 = 271199012+ 29 gt 0
(31)
Therefore Lemma 1(6) follows from (30) and (31)
Lemma 2 Suppose that ℎ(119905) = (6119901+1)1199056119901+6
minus31199056119901+4
minus2(3119901+
2)1199056119901+3
+(3119901+1)1199053119901+6
minus31199011199053119901+5
minus3(119901+1)1199053119901+4
+3(119901+1)1199053119901+2
+31199011199053119901+1
minus (3119901 + 1)1199053119901
+ 2(3119901 + 2)1199053
+ 31199052minus 6119901 minus 1 If 119901 isin
(minus13 minus16) then ℎ(119905) gt 0 for 119905 gt 1
Proof Let ℎ1(119905) = ℎ1015840(119905)119905 ℎ2(119905) = 119905
4minus3119901ℎ10158401015840
1(119905) ℎ3(119905) = ℎ
101584010158401015840
2(119905)119905
and ℎ4(119905) = 1199054minus3119901
ℎ101584010158401015840
3(119905)[9119901(119901 + 1)(3119901 + 1)(3119901 + 2)
2(6119901 + 1)]
then elaborated computations lead to
ℎ (1) = 0 (32)
ℎ1 (119905) = 6 (119901 + 1) (6119901 + 1) 1199056119901+4
minus 6 (3119901 + 2) 1199056119901+2
minus 6 (2119901 + 1) (3119901 + 2) 1199056119901+1
+ 3 (119901 + 2) (3119901 + 1) 1199053119901+4
minus 3119901 (3119901 + 5) 1199053119901+3
minus 3 (119901 + 1) (3119901 + 4) 1199053119901+2
+ 3 (119901 + 1) (3119901 + 2) 1199053119901
+ 3119901 (3119901 + 1) 1199053119901minus1
minus 3119901 (3119901 + 1) 1199053119901minus2
+ 6 (3119901 + 2) 119905 + 6
(33)
ℎ1 (1) = 0 (34)
ℎ1015840
1(119905) = 12 (119901 + 1) (3119901 + 2) (6119901 + 1) 119905
6119901+3
minus 12 (3119901 + 1) (3119901 + 2) 1199056119901+1
minus 6 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199056119901
+ 3 (119901 + 2) (3119901 + 1) (3119901 + 4) 1199053119901+3
minus 9119901 (119901 + 1) (3119901 + 5) 1199053119901+2
minus 3 (119901 + 1) (3119901 + 2) (3119901 + 4) 1199053119901+1
+ 9119901 (119901 + 1) (3119901 + 2) 1199053119901minus1
+ 3119901 (3119901 minus 1) (3119901 + 1) 1199053119901minus2
minus 3119901 (3119901 minus 2) (3119901 + 1) 1199053119901minus3
+ 6 (3119901 + 2)
(35)
4 Chinese Journal of Mathematics
ℎ1015840
1(1) = 0 (36)
ℎ2 (119905) = 36 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+6
minus 12 (3119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+4
minus 36119901 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+3
+ 9 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199056
minus 9119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199055
minus 3 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199054
+ 9119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 1199052
+ 3119901 (3119901 minus 2) (3119901 minus 1) (3119901 + 1) 119905
minus 9119901 (119901 minus 1) (3119901 minus 2) (3119901 + 1)
(37)
ℎ2 (1) = 12 (3119901 + 1) (181199012+ 21119901 + 8) (38)
ℎ1015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2) (6119901 + 1) 119905
3119901+5
minus 12 (3119901 + 1) (3119901 + 2) (3119901 + 4) (6119901 + 1) 1199053119901+3
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+2
+ 54 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199055
minus 45119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199054
minus 12 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199053
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 119905
+ 3119901 (3119901 minus 1) (3119901 minus 2) (3119901 + 1)
(39)
ℎ1015840
2(1) = 12 (3119901 + 1) (9119901 + 7) (18119901
2+ 21119901 + 8) (40)
ℎ10158401015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1)
times (3119901 + 2) (3119901 + 5) (6119901 + 1) 1199053119901+4
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)
times (3119901 + 4) (6119901 + 1) 1199053119901+2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901+1
+ 270 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199054
minus 180119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199053
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199052
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2)
(41)
ℎ10158401015840
2(1) = 36 (3119901 + 1) (119901 + 1) (270119901
3+ 567119901
2+ 384119901 + 104)
(42)
ℎ3 (119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+2
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus1
+ 1080 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 1199052
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 119905
minus 72 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4)
(43)
ℎ3 (1) = 36 (3119901 + 1) (119901 + 1)
times (11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448)
(44)
ℎ1015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
2
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+1
minus 108119901 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901minus1
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus2
+ 2160 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 119905
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5)
(45)
ℎ1015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (14581199015+ 4941119901
4+ 6903119901
3
+48091199012+ 1714119901 + 320)
(46)
ℎ10158401015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 1)
times (3119901 + 2)2(3119901 + 4) (3119901 + 5) (6119901 + 1) 119905
3119901
minus 108119901 (119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(3119901 + 4)
times (6119901 + 1) 1199053119901minus2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 2)
times (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901minus3
+ 2160 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4)
(47)
ℎ10158401015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+96721199012+ 2264119901 + 320)
(48)
Chinese Journal of Mathematics 5
ℎ4 (119905) = 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 1199053
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
gt 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 119905
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
= 24 (3119901 + 4) (91199013+ 33119901
2+ 51119901 + 14) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
(49)
It follows from Lemma 1(6) and (49) that
ℎ4 (119905) gt 0 (50)
for 119905 gt 1From Lemma 1(1)ndash(5) and (38)ndash(48) we clearly see that
ℎ2 (1) gt 0
ℎ1015840
2(1) gt 0
ℎ10158401015840
2(1) gt 0
ℎ3 (1) gt 0
ℎ1015840
3(1) gt 0
ℎ10158401015840
3(1) gt 0
(51)
Therefore Lemma 2 follows from 9119901(119901 + 1)(3119901 + 1)(3119901 +
2)2(6119901+1) gt 0 and (32)ndash(36) together with (50) and (51)
Lemma 3 Inequality [11987223(119886 119887)]120572[119871minus13(119886 119887)]
1minus120572gt 119871 (120572minus2)6
(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
Proof Without loss of generality we assume that 119886 gt 119887 Let119905 = (119886119887)
13gt 1 and 119901 = (120572 minus 2)6 then 119901 isin (minus13 minus16)
120572 = 6119901 + 2 and from (1) and (4) we have
[11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
= (119887 [(1199053119901
+ 1) (1199052+ 1)
3119901+2
(119905 + 1)6119901+1
minus29119901+3
1199056119901+1
(1199053119901+3
+ 1) ])
times (29119901+3
1199056119901+1
(1199053119901
+ 1))
minus1
(52)
Let
119891 (119905) = log (1199053119901
+ 1) + (3119901 + 2) log (1199052+ 1)
+ (6119901 + 1) log (119905 + 1) minus (6119901 + 1) log 119905
minus log (1199053119901+3
+ 1) minus 3 (3119901 + 1) log 2
(53)
Then simple computations lead to
119891 (1) = 0 (54)
1198911015840(119905) =
ℎ (119905)
119905 (119905 + 1) (1199052+ 1) (119905
3119901+ 1) (119905
3119901+3+ 1)
(55)
where ℎ(119905) is defined as in Lemma 2From (54) and (55) together with Lemma 2 we clearly see
that
119891 (119905) gt 0 (56)
for 119905 gt 1Therefore Lemma 3 follows from (52) and (53) together
with (56)
3 Main Results
Theorem 4 Inequality 119871 (120572minus2)6(119886 119887) lt 119868120572(119886 119887)119871
1minus120572(119886 119887) lt
1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possible lower andupper Lehmer mean bounds for the product 119868120572(119886 119887)1198711minus120572(119886 119887)
Proof Inequality 119868120572(119886 119887)119871
1minus120572(119886 119887) lt 1198710(119886 119887) follows di-
rectly from (10) and (11)For the other inequality we note that
119868120572(119886 119887) 119871
1minus120572(119886 119887) minus 119871 (120572minus2)6 (119886 119887)
= 119868120572(119886 119887) 119871
1minus120572(119886 119887) minus [11987223 (119886 119887)]
120572[119871minus13 (119886 119887)]
1minus120572
+ [11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
(57)
Therefore 119868120572(119886 119887)1198711minus120572(119886 119887) gt 119871 (120572minus2)6(119886 119887) follows from(7) and (10) together with Lemma 3 and (57)
Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are thebest possible lower and upper Lehmer mean bounds for theproduct 119868120572(119886 119887)1198711minus120572(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
119868120572(1 119909) 119871
1minus120572(1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
119890120572((log119909)(119909minus1)minus1)
times(1 minus
1
119909
)
1minus120572119909120576
(log119909)1minus120572
] = +infin
119871 (120572minus2)6+120576 (1 1 + 119909) minus 119868120572(1 1 + 119909) 119871
1minus120572(1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
(58)
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Chinese Journal of Mathematics
of two positive numbers 119886 and 119887 respectively Then it is wellknown that
min 119886 119887 lt 119867 (119886 119887) = 119871minus1 (119886 119887)
= 119872minus1 (119886 119887) lt 119866 (119886 119887) = 119871minus12 (119886 119887)
= 1198720 (119886 119887) lt 119871 (119886 119887) lt 119868 (119886 119887) lt 119860 (119886 119887)
= 1198710 (119886 119887) = 1198721 (119886 119887) lt max 119886 119887
(6)
for all 119886 119887 gt 0 with 119886 = 119887The following sharp bounds for 119871 119868 (119871119868)12 and (119871+119868)2
in terms of power means are proved in [2ndash4 6 8 9 33]
11987223 (119886 119887) lt 119868 (119886 119887) lt 119872log 2 (119886 119887) (7)
1198720 (119886 119887) lt 119871 (119886 119887) lt 11987213 (119886 119887)
1198720 (119886 119887) lt radic119868 (119886 119887) 119871 (119886 119887) lt 11987212 (119886 119887)
1
2
(119868 (119886 119887) + 119871 (119886 119887)) lt 11987212 (119886 119887)
(8)
for all 119886 119887 gt 0 with 119886 = 119887Alzer and Qiu [19] proved that
120572119860 (119886 119887) + (1 minus 120572)119866 (119886 119887)
lt 119868 (119886 119887) lt 120573119860 (119886 119887) + (1 minus 120573)119866 (119886 119887)
119872119901 (119886 119887) lt
1
2
(119868 (119886 119887) + 119871 (119886 119887))
(9)
for all 119886 119887 gt 0 with 119886 = 119887 if and only if 120572 le 23 120573 ge 2119890 =
073575 and 119901 le log 2(1 + log 2) = 040938 The following sharp upper and lower Lehmer mean
bounds for 119871 119868 (119871119868)12 and (119871 + 119868)2 are presented in [34]
119871minus13 (119886 119887) lt 119871 (119886 119887) lt 1198710 (119886 119887) (10)
119872minus16 (119886 119887) lt 119868 (119886 119887) lt 1198710 (119886 119887) (11)
119871minus14 (119886 119887) lt 11986812
(119886 119887) 11987112
(119886 119887) lt 1198710 (119886 119887)
119871minus14 (119886 119887) lt
1
2
(119868 (119886 119887) + 119871 (119886 119887)) lt 1198710 (119886 119887)
(12)
for all 119886 119887 gt 0 with 119886 = 119887The purpose of this paper is to present the best possible
upper and lower Lehmer mean bounds of the product119868120572(119886 119887)119871
1minus120572(119886 119887) and the sum 120572119868(119886 119887)+ (1minus120572)119871(119886 119887) for any
120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
2 Lemmas
In order to establish ourmain result we need several lemmaswhich we present in this section
Lemma 1 If 119901 ge minus13 then the following statements are true
(1) 181199012+ 21119901 + 8 gt 0
(2) 2701199013+ 567119901
2+ 384119901 + 104 gt 0
(3) 11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448 gt 0
(4) 14581199015+4941119901
4+6903119901
3+4809119901
2+1714119901+320 gt 0
(5) 53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3+ 9672119901
2+
2264119901 + 320 gt 0
(6) 91199013+ 33119901
2+ 51119901 + 14 gt 0
Proof (1) We clearly see that
181199012+ 21119901 + 8 gt 18119901
2+ (minus
1
3
) times 21 + 8
= 181199012+ 1 gt 0
(13)
(2) Let
1198921 (119901) = 2701199013+ 567119901
2+ 384119901 + 104 (14)
Then simple computations lead to
1198921 (minus1
3
) = 29 gt 0
1198921015840
1(119901) = 6 (135119901
2+ 189119901 + 64)
gt 6 [1351199012+ 189 times (minus
1
3
) + 64]
= 6 (1351199012+ 1) gt 0
(15)
Therefore Lemma 1(2) follows from (14) and (15)(3) Let
1198922 (119901) = 11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448 (16)
Then simple computations yield
1198922 (minus1
3
) = 116 gt 0
1198921015840
2(119901) = 4536119901
3+ 10287119901
2+ 7758119901 + 1950
1198921015840
2(minus
1
3
) = 339 gt 0
11989210158401015840
2(119901) = 13608119901
2+ 20574119901 + 7758
gt 136081199012+ 20574 times (minus
1
3
) + 7758
= 136081199012+ 900 gt 0
(17)
Therefore Lemma 1(3) follows from (16) and (17)(4) Let
1198923 (119901) = 14581199015+ 4941119901
4+ 6903119901
3
+ 48091199012+ 1714119901 + 320
(18)
Chinese Journal of Mathematics 3
Then simple computations lead to
1198923 (minus1
3
) =
247
3
gt 0
1198921015840
3(119901) = 7290119901
4+ 19764119901
3+ 20709119901
2
+ 9618119901 + 1714
1198921015840
3(minus
1
3
) = 167 gt 0
11989210158401015840
3(119901) = 29160119901
3+ 59292119901
2+ 41418119901 + 9618
11989210158401015840
3(minus
1
3
) = 1320 gt 0
119892101584010158401015840
3(119901) = 87480119901
2+ 118584119901 + 41418
gt 874801199012+ 41418 + 118584 times (minus
1
3
)
= 874801199012+ 1890 gt 0
(19)
Therefore Lemma 1(4) follows from (18) and (19)(5) Let
1198924 (119901) = 53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+ 96721199012+ 2264119901 + 320
(20)
Then simple computations yield
1198924 (minus1
3
) =
284
3
gt 0 (21)
1198921015840
4(119901) = 32076119901
5+ 91125119901
4+ 109188119901
3
+ 659881199012+ 19344119901 + 2264
(22)
1198921015840
4(minus
1
3
) = 97 gt 0 (23)
11989210158401015840
4(119901) = 160380119901
4+ 364500119901
3
+ 3275641199012+ 131976119901 + 19344
(24)
11989210158401015840
4(minus
1
3
) = 228 gt 0 (25)
119892101584010158401015840
4(119901) = 108 (5940119901
3+ 10125119901
2+ 6066119901 + 1222) (26)
119892101584010158401015840
4(minus
1
3
) = 11340 gt 0 (27)
119892(4)
4(119901) = 1944 (990119901
2+ 1125119901 + 337) (28)
It follows from (28) and the discriminant Δ = (1125)2minus
4 times 990 times 337 = minus68895 lt 0 of the quadratic function 119891(119901) =
9901199012+ 1125119901 + 337 that
119892(4)
4(119901) gt 0 (29)
Therefore Lemma 1(5) follows from (20)ndash(27) and (29)
(6) Let
1198925 (119901) = 91199013+ 33119901
2+ 51119901 + 14 (30)
Then we have
1198925 (minus1
3
) =
1
3
gt 0
1198921015840
5(119901) = 27119901
2+ 66119901 + 51
gt 271199012minus 66 times (
1
3
) + 51 = 271199012+ 29 gt 0
(31)
Therefore Lemma 1(6) follows from (30) and (31)
Lemma 2 Suppose that ℎ(119905) = (6119901+1)1199056119901+6
minus31199056119901+4
minus2(3119901+
2)1199056119901+3
+(3119901+1)1199053119901+6
minus31199011199053119901+5
minus3(119901+1)1199053119901+4
+3(119901+1)1199053119901+2
+31199011199053119901+1
minus (3119901 + 1)1199053119901
+ 2(3119901 + 2)1199053
+ 31199052minus 6119901 minus 1 If 119901 isin
(minus13 minus16) then ℎ(119905) gt 0 for 119905 gt 1
Proof Let ℎ1(119905) = ℎ1015840(119905)119905 ℎ2(119905) = 119905
4minus3119901ℎ10158401015840
1(119905) ℎ3(119905) = ℎ
101584010158401015840
2(119905)119905
and ℎ4(119905) = 1199054minus3119901
ℎ101584010158401015840
3(119905)[9119901(119901 + 1)(3119901 + 1)(3119901 + 2)
2(6119901 + 1)]
then elaborated computations lead to
ℎ (1) = 0 (32)
ℎ1 (119905) = 6 (119901 + 1) (6119901 + 1) 1199056119901+4
minus 6 (3119901 + 2) 1199056119901+2
minus 6 (2119901 + 1) (3119901 + 2) 1199056119901+1
+ 3 (119901 + 2) (3119901 + 1) 1199053119901+4
minus 3119901 (3119901 + 5) 1199053119901+3
minus 3 (119901 + 1) (3119901 + 4) 1199053119901+2
+ 3 (119901 + 1) (3119901 + 2) 1199053119901
+ 3119901 (3119901 + 1) 1199053119901minus1
minus 3119901 (3119901 + 1) 1199053119901minus2
+ 6 (3119901 + 2) 119905 + 6
(33)
ℎ1 (1) = 0 (34)
ℎ1015840
1(119905) = 12 (119901 + 1) (3119901 + 2) (6119901 + 1) 119905
6119901+3
minus 12 (3119901 + 1) (3119901 + 2) 1199056119901+1
minus 6 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199056119901
+ 3 (119901 + 2) (3119901 + 1) (3119901 + 4) 1199053119901+3
minus 9119901 (119901 + 1) (3119901 + 5) 1199053119901+2
minus 3 (119901 + 1) (3119901 + 2) (3119901 + 4) 1199053119901+1
+ 9119901 (119901 + 1) (3119901 + 2) 1199053119901minus1
+ 3119901 (3119901 minus 1) (3119901 + 1) 1199053119901minus2
minus 3119901 (3119901 minus 2) (3119901 + 1) 1199053119901minus3
+ 6 (3119901 + 2)
(35)
4 Chinese Journal of Mathematics
ℎ1015840
1(1) = 0 (36)
ℎ2 (119905) = 36 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+6
minus 12 (3119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+4
minus 36119901 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+3
+ 9 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199056
minus 9119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199055
minus 3 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199054
+ 9119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 1199052
+ 3119901 (3119901 minus 2) (3119901 minus 1) (3119901 + 1) 119905
minus 9119901 (119901 minus 1) (3119901 minus 2) (3119901 + 1)
(37)
ℎ2 (1) = 12 (3119901 + 1) (181199012+ 21119901 + 8) (38)
ℎ1015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2) (6119901 + 1) 119905
3119901+5
minus 12 (3119901 + 1) (3119901 + 2) (3119901 + 4) (6119901 + 1) 1199053119901+3
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+2
+ 54 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199055
minus 45119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199054
minus 12 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199053
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 119905
+ 3119901 (3119901 minus 1) (3119901 minus 2) (3119901 + 1)
(39)
ℎ1015840
2(1) = 12 (3119901 + 1) (9119901 + 7) (18119901
2+ 21119901 + 8) (40)
ℎ10158401015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1)
times (3119901 + 2) (3119901 + 5) (6119901 + 1) 1199053119901+4
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)
times (3119901 + 4) (6119901 + 1) 1199053119901+2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901+1
+ 270 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199054
minus 180119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199053
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199052
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2)
(41)
ℎ10158401015840
2(1) = 36 (3119901 + 1) (119901 + 1) (270119901
3+ 567119901
2+ 384119901 + 104)
(42)
ℎ3 (119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+2
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus1
+ 1080 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 1199052
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 119905
minus 72 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4)
(43)
ℎ3 (1) = 36 (3119901 + 1) (119901 + 1)
times (11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448)
(44)
ℎ1015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
2
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+1
minus 108119901 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901minus1
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus2
+ 2160 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 119905
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5)
(45)
ℎ1015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (14581199015+ 4941119901
4+ 6903119901
3
+48091199012+ 1714119901 + 320)
(46)
ℎ10158401015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 1)
times (3119901 + 2)2(3119901 + 4) (3119901 + 5) (6119901 + 1) 119905
3119901
minus 108119901 (119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(3119901 + 4)
times (6119901 + 1) 1199053119901minus2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 2)
times (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901minus3
+ 2160 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4)
(47)
ℎ10158401015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+96721199012+ 2264119901 + 320)
(48)
Chinese Journal of Mathematics 5
ℎ4 (119905) = 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 1199053
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
gt 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 119905
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
= 24 (3119901 + 4) (91199013+ 33119901
2+ 51119901 + 14) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
(49)
It follows from Lemma 1(6) and (49) that
ℎ4 (119905) gt 0 (50)
for 119905 gt 1From Lemma 1(1)ndash(5) and (38)ndash(48) we clearly see that
ℎ2 (1) gt 0
ℎ1015840
2(1) gt 0
ℎ10158401015840
2(1) gt 0
ℎ3 (1) gt 0
ℎ1015840
3(1) gt 0
ℎ10158401015840
3(1) gt 0
(51)
Therefore Lemma 2 follows from 9119901(119901 + 1)(3119901 + 1)(3119901 +
2)2(6119901+1) gt 0 and (32)ndash(36) together with (50) and (51)
Lemma 3 Inequality [11987223(119886 119887)]120572[119871minus13(119886 119887)]
1minus120572gt 119871 (120572minus2)6
(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
Proof Without loss of generality we assume that 119886 gt 119887 Let119905 = (119886119887)
13gt 1 and 119901 = (120572 minus 2)6 then 119901 isin (minus13 minus16)
120572 = 6119901 + 2 and from (1) and (4) we have
[11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
= (119887 [(1199053119901
+ 1) (1199052+ 1)
3119901+2
(119905 + 1)6119901+1
minus29119901+3
1199056119901+1
(1199053119901+3
+ 1) ])
times (29119901+3
1199056119901+1
(1199053119901
+ 1))
minus1
(52)
Let
119891 (119905) = log (1199053119901
+ 1) + (3119901 + 2) log (1199052+ 1)
+ (6119901 + 1) log (119905 + 1) minus (6119901 + 1) log 119905
minus log (1199053119901+3
+ 1) minus 3 (3119901 + 1) log 2
(53)
Then simple computations lead to
119891 (1) = 0 (54)
1198911015840(119905) =
ℎ (119905)
119905 (119905 + 1) (1199052+ 1) (119905
3119901+ 1) (119905
3119901+3+ 1)
(55)
where ℎ(119905) is defined as in Lemma 2From (54) and (55) together with Lemma 2 we clearly see
that
119891 (119905) gt 0 (56)
for 119905 gt 1Therefore Lemma 3 follows from (52) and (53) together
with (56)
3 Main Results
Theorem 4 Inequality 119871 (120572minus2)6(119886 119887) lt 119868120572(119886 119887)119871
1minus120572(119886 119887) lt
1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possible lower andupper Lehmer mean bounds for the product 119868120572(119886 119887)1198711minus120572(119886 119887)
Proof Inequality 119868120572(119886 119887)119871
1minus120572(119886 119887) lt 1198710(119886 119887) follows di-
rectly from (10) and (11)For the other inequality we note that
119868120572(119886 119887) 119871
1minus120572(119886 119887) minus 119871 (120572minus2)6 (119886 119887)
= 119868120572(119886 119887) 119871
1minus120572(119886 119887) minus [11987223 (119886 119887)]
120572[119871minus13 (119886 119887)]
1minus120572
+ [11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
(57)
Therefore 119868120572(119886 119887)1198711minus120572(119886 119887) gt 119871 (120572minus2)6(119886 119887) follows from(7) and (10) together with Lemma 3 and (57)
Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are thebest possible lower and upper Lehmer mean bounds for theproduct 119868120572(119886 119887)1198711minus120572(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
119868120572(1 119909) 119871
1minus120572(1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
119890120572((log119909)(119909minus1)minus1)
times(1 minus
1
119909
)
1minus120572119909120576
(log119909)1minus120572
] = +infin
119871 (120572minus2)6+120576 (1 1 + 119909) minus 119868120572(1 1 + 119909) 119871
1minus120572(1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
(58)
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Chinese Journal of Mathematics 3
Then simple computations lead to
1198923 (minus1
3
) =
247
3
gt 0
1198921015840
3(119901) = 7290119901
4+ 19764119901
3+ 20709119901
2
+ 9618119901 + 1714
1198921015840
3(minus
1
3
) = 167 gt 0
11989210158401015840
3(119901) = 29160119901
3+ 59292119901
2+ 41418119901 + 9618
11989210158401015840
3(minus
1
3
) = 1320 gt 0
119892101584010158401015840
3(119901) = 87480119901
2+ 118584119901 + 41418
gt 874801199012+ 41418 + 118584 times (minus
1
3
)
= 874801199012+ 1890 gt 0
(19)
Therefore Lemma 1(4) follows from (18) and (19)(5) Let
1198924 (119901) = 53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+ 96721199012+ 2264119901 + 320
(20)
Then simple computations yield
1198924 (minus1
3
) =
284
3
gt 0 (21)
1198921015840
4(119901) = 32076119901
5+ 91125119901
4+ 109188119901
3
+ 659881199012+ 19344119901 + 2264
(22)
1198921015840
4(minus
1
3
) = 97 gt 0 (23)
11989210158401015840
4(119901) = 160380119901
4+ 364500119901
3
+ 3275641199012+ 131976119901 + 19344
(24)
11989210158401015840
4(minus
1
3
) = 228 gt 0 (25)
119892101584010158401015840
4(119901) = 108 (5940119901
3+ 10125119901
2+ 6066119901 + 1222) (26)
119892101584010158401015840
4(minus
1
3
) = 11340 gt 0 (27)
119892(4)
4(119901) = 1944 (990119901
2+ 1125119901 + 337) (28)
It follows from (28) and the discriminant Δ = (1125)2minus
4 times 990 times 337 = minus68895 lt 0 of the quadratic function 119891(119901) =
9901199012+ 1125119901 + 337 that
119892(4)
4(119901) gt 0 (29)
Therefore Lemma 1(5) follows from (20)ndash(27) and (29)
(6) Let
1198925 (119901) = 91199013+ 33119901
2+ 51119901 + 14 (30)
Then we have
1198925 (minus1
3
) =
1
3
gt 0
1198921015840
5(119901) = 27119901
2+ 66119901 + 51
gt 271199012minus 66 times (
1
3
) + 51 = 271199012+ 29 gt 0
(31)
Therefore Lemma 1(6) follows from (30) and (31)
Lemma 2 Suppose that ℎ(119905) = (6119901+1)1199056119901+6
minus31199056119901+4
minus2(3119901+
2)1199056119901+3
+(3119901+1)1199053119901+6
minus31199011199053119901+5
minus3(119901+1)1199053119901+4
+3(119901+1)1199053119901+2
+31199011199053119901+1
minus (3119901 + 1)1199053119901
+ 2(3119901 + 2)1199053
+ 31199052minus 6119901 minus 1 If 119901 isin
(minus13 minus16) then ℎ(119905) gt 0 for 119905 gt 1
Proof Let ℎ1(119905) = ℎ1015840(119905)119905 ℎ2(119905) = 119905
4minus3119901ℎ10158401015840
1(119905) ℎ3(119905) = ℎ
101584010158401015840
2(119905)119905
and ℎ4(119905) = 1199054minus3119901
ℎ101584010158401015840
3(119905)[9119901(119901 + 1)(3119901 + 1)(3119901 + 2)
2(6119901 + 1)]
then elaborated computations lead to
ℎ (1) = 0 (32)
ℎ1 (119905) = 6 (119901 + 1) (6119901 + 1) 1199056119901+4
minus 6 (3119901 + 2) 1199056119901+2
minus 6 (2119901 + 1) (3119901 + 2) 1199056119901+1
+ 3 (119901 + 2) (3119901 + 1) 1199053119901+4
minus 3119901 (3119901 + 5) 1199053119901+3
minus 3 (119901 + 1) (3119901 + 4) 1199053119901+2
+ 3 (119901 + 1) (3119901 + 2) 1199053119901
+ 3119901 (3119901 + 1) 1199053119901minus1
minus 3119901 (3119901 + 1) 1199053119901minus2
+ 6 (3119901 + 2) 119905 + 6
(33)
ℎ1 (1) = 0 (34)
ℎ1015840
1(119905) = 12 (119901 + 1) (3119901 + 2) (6119901 + 1) 119905
6119901+3
minus 12 (3119901 + 1) (3119901 + 2) 1199056119901+1
minus 6 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199056119901
+ 3 (119901 + 2) (3119901 + 1) (3119901 + 4) 1199053119901+3
minus 9119901 (119901 + 1) (3119901 + 5) 1199053119901+2
minus 3 (119901 + 1) (3119901 + 2) (3119901 + 4) 1199053119901+1
+ 9119901 (119901 + 1) (3119901 + 2) 1199053119901minus1
+ 3119901 (3119901 minus 1) (3119901 + 1) 1199053119901minus2
minus 3119901 (3119901 minus 2) (3119901 + 1) 1199053119901minus3
+ 6 (3119901 + 2)
(35)
4 Chinese Journal of Mathematics
ℎ1015840
1(1) = 0 (36)
ℎ2 (119905) = 36 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+6
minus 12 (3119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+4
minus 36119901 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+3
+ 9 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199056
minus 9119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199055
minus 3 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199054
+ 9119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 1199052
+ 3119901 (3119901 minus 2) (3119901 minus 1) (3119901 + 1) 119905
minus 9119901 (119901 minus 1) (3119901 minus 2) (3119901 + 1)
(37)
ℎ2 (1) = 12 (3119901 + 1) (181199012+ 21119901 + 8) (38)
ℎ1015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2) (6119901 + 1) 119905
3119901+5
minus 12 (3119901 + 1) (3119901 + 2) (3119901 + 4) (6119901 + 1) 1199053119901+3
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+2
+ 54 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199055
minus 45119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199054
minus 12 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199053
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 119905
+ 3119901 (3119901 minus 1) (3119901 minus 2) (3119901 + 1)
(39)
ℎ1015840
2(1) = 12 (3119901 + 1) (9119901 + 7) (18119901
2+ 21119901 + 8) (40)
ℎ10158401015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1)
times (3119901 + 2) (3119901 + 5) (6119901 + 1) 1199053119901+4
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)
times (3119901 + 4) (6119901 + 1) 1199053119901+2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901+1
+ 270 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199054
minus 180119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199053
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199052
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2)
(41)
ℎ10158401015840
2(1) = 36 (3119901 + 1) (119901 + 1) (270119901
3+ 567119901
2+ 384119901 + 104)
(42)
ℎ3 (119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+2
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus1
+ 1080 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 1199052
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 119905
minus 72 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4)
(43)
ℎ3 (1) = 36 (3119901 + 1) (119901 + 1)
times (11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448)
(44)
ℎ1015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
2
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+1
minus 108119901 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901minus1
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus2
+ 2160 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 119905
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5)
(45)
ℎ1015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (14581199015+ 4941119901
4+ 6903119901
3
+48091199012+ 1714119901 + 320)
(46)
ℎ10158401015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 1)
times (3119901 + 2)2(3119901 + 4) (3119901 + 5) (6119901 + 1) 119905
3119901
minus 108119901 (119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(3119901 + 4)
times (6119901 + 1) 1199053119901minus2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 2)
times (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901minus3
+ 2160 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4)
(47)
ℎ10158401015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+96721199012+ 2264119901 + 320)
(48)
Chinese Journal of Mathematics 5
ℎ4 (119905) = 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 1199053
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
gt 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 119905
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
= 24 (3119901 + 4) (91199013+ 33119901
2+ 51119901 + 14) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
(49)
It follows from Lemma 1(6) and (49) that
ℎ4 (119905) gt 0 (50)
for 119905 gt 1From Lemma 1(1)ndash(5) and (38)ndash(48) we clearly see that
ℎ2 (1) gt 0
ℎ1015840
2(1) gt 0
ℎ10158401015840
2(1) gt 0
ℎ3 (1) gt 0
ℎ1015840
3(1) gt 0
ℎ10158401015840
3(1) gt 0
(51)
Therefore Lemma 2 follows from 9119901(119901 + 1)(3119901 + 1)(3119901 +
2)2(6119901+1) gt 0 and (32)ndash(36) together with (50) and (51)
Lemma 3 Inequality [11987223(119886 119887)]120572[119871minus13(119886 119887)]
1minus120572gt 119871 (120572minus2)6
(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
Proof Without loss of generality we assume that 119886 gt 119887 Let119905 = (119886119887)
13gt 1 and 119901 = (120572 minus 2)6 then 119901 isin (minus13 minus16)
120572 = 6119901 + 2 and from (1) and (4) we have
[11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
= (119887 [(1199053119901
+ 1) (1199052+ 1)
3119901+2
(119905 + 1)6119901+1
minus29119901+3
1199056119901+1
(1199053119901+3
+ 1) ])
times (29119901+3
1199056119901+1
(1199053119901
+ 1))
minus1
(52)
Let
119891 (119905) = log (1199053119901
+ 1) + (3119901 + 2) log (1199052+ 1)
+ (6119901 + 1) log (119905 + 1) minus (6119901 + 1) log 119905
minus log (1199053119901+3
+ 1) minus 3 (3119901 + 1) log 2
(53)
Then simple computations lead to
119891 (1) = 0 (54)
1198911015840(119905) =
ℎ (119905)
119905 (119905 + 1) (1199052+ 1) (119905
3119901+ 1) (119905
3119901+3+ 1)
(55)
where ℎ(119905) is defined as in Lemma 2From (54) and (55) together with Lemma 2 we clearly see
that
119891 (119905) gt 0 (56)
for 119905 gt 1Therefore Lemma 3 follows from (52) and (53) together
with (56)
3 Main Results
Theorem 4 Inequality 119871 (120572minus2)6(119886 119887) lt 119868120572(119886 119887)119871
1minus120572(119886 119887) lt
1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possible lower andupper Lehmer mean bounds for the product 119868120572(119886 119887)1198711minus120572(119886 119887)
Proof Inequality 119868120572(119886 119887)119871
1minus120572(119886 119887) lt 1198710(119886 119887) follows di-
rectly from (10) and (11)For the other inequality we note that
119868120572(119886 119887) 119871
1minus120572(119886 119887) minus 119871 (120572minus2)6 (119886 119887)
= 119868120572(119886 119887) 119871
1minus120572(119886 119887) minus [11987223 (119886 119887)]
120572[119871minus13 (119886 119887)]
1minus120572
+ [11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
(57)
Therefore 119868120572(119886 119887)1198711minus120572(119886 119887) gt 119871 (120572minus2)6(119886 119887) follows from(7) and (10) together with Lemma 3 and (57)
Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are thebest possible lower and upper Lehmer mean bounds for theproduct 119868120572(119886 119887)1198711minus120572(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
119868120572(1 119909) 119871
1minus120572(1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
119890120572((log119909)(119909minus1)minus1)
times(1 minus
1
119909
)
1minus120572119909120576
(log119909)1minus120572
] = +infin
119871 (120572minus2)6+120576 (1 1 + 119909) minus 119868120572(1 1 + 119909) 119871
1minus120572(1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
(58)
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Chinese Journal of Mathematics
ℎ1015840
1(1) = 0 (36)
ℎ2 (119905) = 36 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+6
minus 12 (3119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+4
minus 36119901 (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+3
+ 9 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199056
minus 9119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199055
minus 3 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199054
+ 9119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 1199052
+ 3119901 (3119901 minus 2) (3119901 minus 1) (3119901 + 1) 119905
minus 9119901 (119901 minus 1) (3119901 minus 2) (3119901 + 1)
(37)
ℎ2 (1) = 12 (3119901 + 1) (181199012+ 21119901 + 8) (38)
ℎ1015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2) (6119901 + 1) 119905
3119901+5
minus 12 (3119901 + 1) (3119901 + 2) (3119901 + 4) (6119901 + 1) 1199053119901+3
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2) (6119901 + 1) 1199053119901+2
+ 54 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199055
minus 45119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199054
minus 12 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199053
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2) 119905
+ 3119901 (3119901 minus 1) (3119901 minus 2) (3119901 + 1)
(39)
ℎ1015840
2(1) = 12 (3119901 + 1) (9119901 + 7) (18119901
2+ 21119901 + 8) (40)
ℎ10158401015840
2(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1)
times (3119901 + 2) (3119901 + 5) (6119901 + 1) 1199053119901+4
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)
times (3119901 + 4) (6119901 + 1) 1199053119901+2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901+1
+ 270 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4) 1199054
minus 180119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 1199053
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4) 1199052
+ 18119901 (119901 + 1) (3119901 minus 1) (3119901 + 2)
(41)
ℎ10158401015840
2(1) = 36 (3119901 + 1) (119901 + 1) (270119901
3+ 567119901
2+ 384119901 + 104)
(42)
ℎ3 (119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+2
minus 36 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus1
+ 1080 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 1199052
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5) 119905
minus 72 (119901 + 1) (3119901 + 1) (3119901 + 2) (3119901 + 4)
(43)
ℎ3 (1) = 36 (3119901 + 1) (119901 + 1)
times (11341199014+ 3429119901
3+ 3879119901
2+ 1950119901 + 448)
(44)
ℎ1015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 2)
2
times (3119901 + 4) (3119901 + 5) (6119901 + 1) 1199053119901+1
minus 108119901 (119901 + 1) (3119901 + 1) (3119901 + 2)2
times (3119901 + 4) (6119901 + 1) 1199053119901minus1
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2
times (6119901 + 1) 1199053119901minus2
+ 2160 (119901 + 1) (119901 + 2)
times (3119901 + 1) (3119901 + 4) 119905
minus 540119901 (119901 + 1) (3119901 + 2) (3119901 + 5)
(45)
ℎ1015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (14581199015+ 4941119901
4+ 6903119901
3
+48091199012+ 1714119901 + 320)
(46)
ℎ10158401015840
3(119905) = 108 (119901 + 1) (119901 + 2) (2119901 + 1) (3119901 + 1)
times (3119901 + 2)2(3119901 + 4) (3119901 + 5) (6119901 + 1) 119905
3119901
minus 108119901 (119901 + 1) (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(3119901 + 4)
times (6119901 + 1) 1199053119901minus2
minus 108119901 (119901 + 1) (2119901 + 1) (3119901 minus 2)
times (3119901 minus 1) (3119901 + 1) (3119901 + 2)2(6119901 + 1) 119905
3119901minus3
+ 2160 (119901 + 1) (119901 + 2) (3119901 + 1) (3119901 + 4)
(47)
ℎ10158401015840
3(1) = 108 (119901 + 1) (3119901 + 1)
times (53461199016+ 18225119901
5+ 27297119901
4+ 21996119901
3
+96721199012+ 2264119901 + 320)
(48)
Chinese Journal of Mathematics 5
ℎ4 (119905) = 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 1199053
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
gt 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 119905
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
= 24 (3119901 + 4) (91199013+ 33119901
2+ 51119901 + 14) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
(49)
It follows from Lemma 1(6) and (49) that
ℎ4 (119905) gt 0 (50)
for 119905 gt 1From Lemma 1(1)ndash(5) and (38)ndash(48) we clearly see that
ℎ2 (1) gt 0
ℎ1015840
2(1) gt 0
ℎ10158401015840
2(1) gt 0
ℎ3 (1) gt 0
ℎ1015840
3(1) gt 0
ℎ10158401015840
3(1) gt 0
(51)
Therefore Lemma 2 follows from 9119901(119901 + 1)(3119901 + 1)(3119901 +
2)2(6119901+1) gt 0 and (32)ndash(36) together with (50) and (51)
Lemma 3 Inequality [11987223(119886 119887)]120572[119871minus13(119886 119887)]
1minus120572gt 119871 (120572minus2)6
(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
Proof Without loss of generality we assume that 119886 gt 119887 Let119905 = (119886119887)
13gt 1 and 119901 = (120572 minus 2)6 then 119901 isin (minus13 minus16)
120572 = 6119901 + 2 and from (1) and (4) we have
[11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
= (119887 [(1199053119901
+ 1) (1199052+ 1)
3119901+2
(119905 + 1)6119901+1
minus29119901+3
1199056119901+1
(1199053119901+3
+ 1) ])
times (29119901+3
1199056119901+1
(1199053119901
+ 1))
minus1
(52)
Let
119891 (119905) = log (1199053119901
+ 1) + (3119901 + 2) log (1199052+ 1)
+ (6119901 + 1) log (119905 + 1) minus (6119901 + 1) log 119905
minus log (1199053119901+3
+ 1) minus 3 (3119901 + 1) log 2
(53)
Then simple computations lead to
119891 (1) = 0 (54)
1198911015840(119905) =
ℎ (119905)
119905 (119905 + 1) (1199052+ 1) (119905
3119901+ 1) (119905
3119901+3+ 1)
(55)
where ℎ(119905) is defined as in Lemma 2From (54) and (55) together with Lemma 2 we clearly see
that
119891 (119905) gt 0 (56)
for 119905 gt 1Therefore Lemma 3 follows from (52) and (53) together
with (56)
3 Main Results
Theorem 4 Inequality 119871 (120572minus2)6(119886 119887) lt 119868120572(119886 119887)119871
1minus120572(119886 119887) lt
1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possible lower andupper Lehmer mean bounds for the product 119868120572(119886 119887)1198711minus120572(119886 119887)
Proof Inequality 119868120572(119886 119887)119871
1minus120572(119886 119887) lt 1198710(119886 119887) follows di-
rectly from (10) and (11)For the other inequality we note that
119868120572(119886 119887) 119871
1minus120572(119886 119887) minus 119871 (120572minus2)6 (119886 119887)
= 119868120572(119886 119887) 119871
1minus120572(119886 119887) minus [11987223 (119886 119887)]
120572[119871minus13 (119886 119887)]
1minus120572
+ [11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
(57)
Therefore 119868120572(119886 119887)1198711minus120572(119886 119887) gt 119871 (120572minus2)6(119886 119887) follows from(7) and (10) together with Lemma 3 and (57)
Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are thebest possible lower and upper Lehmer mean bounds for theproduct 119868120572(119886 119887)1198711minus120572(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
119868120572(1 119909) 119871
1minus120572(1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
119890120572((log119909)(119909minus1)minus1)
times(1 minus
1
119909
)
1minus120572119909120576
(log119909)1minus120572
] = +infin
119871 (120572minus2)6+120576 (1 1 + 119909) minus 119868120572(1 1 + 119909) 119871
1minus120572(1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
(58)
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 5
ℎ4 (119905) = 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 1199053
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
gt 36 (119901 + 2) (2119901 + 1) (3119901 + 4) (3119901 + 5) 119905
minus 12 (1 minus 3119901) (2 minus 3119901) (3119901 + 4) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
= 24 (3119901 + 4) (91199013+ 33119901
2+ 51119901 + 14) 119905
+ 36 (1 minus 119901) (1 minus 3119901) (2 minus 3119901) (2119901 + 1)
(49)
It follows from Lemma 1(6) and (49) that
ℎ4 (119905) gt 0 (50)
for 119905 gt 1From Lemma 1(1)ndash(5) and (38)ndash(48) we clearly see that
ℎ2 (1) gt 0
ℎ1015840
2(1) gt 0
ℎ10158401015840
2(1) gt 0
ℎ3 (1) gt 0
ℎ1015840
3(1) gt 0
ℎ10158401015840
3(1) gt 0
(51)
Therefore Lemma 2 follows from 9119901(119901 + 1)(3119901 + 1)(3119901 +
2)2(6119901+1) gt 0 and (32)ndash(36) together with (50) and (51)
Lemma 3 Inequality [11987223(119886 119887)]120572[119871minus13(119886 119887)]
1minus120572gt 119871 (120572minus2)6
(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887
Proof Without loss of generality we assume that 119886 gt 119887 Let119905 = (119886119887)
13gt 1 and 119901 = (120572 minus 2)6 then 119901 isin (minus13 minus16)
120572 = 6119901 + 2 and from (1) and (4) we have
[11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
= (119887 [(1199053119901
+ 1) (1199052+ 1)
3119901+2
(119905 + 1)6119901+1
minus29119901+3
1199056119901+1
(1199053119901+3
+ 1) ])
times (29119901+3
1199056119901+1
(1199053119901
+ 1))
minus1
(52)
Let
119891 (119905) = log (1199053119901
+ 1) + (3119901 + 2) log (1199052+ 1)
+ (6119901 + 1) log (119905 + 1) minus (6119901 + 1) log 119905
minus log (1199053119901+3
+ 1) minus 3 (3119901 + 1) log 2
(53)
Then simple computations lead to
119891 (1) = 0 (54)
1198911015840(119905) =
ℎ (119905)
119905 (119905 + 1) (1199052+ 1) (119905
3119901+ 1) (119905
3119901+3+ 1)
(55)
where ℎ(119905) is defined as in Lemma 2From (54) and (55) together with Lemma 2 we clearly see
that
119891 (119905) gt 0 (56)
for 119905 gt 1Therefore Lemma 3 follows from (52) and (53) together
with (56)
3 Main Results
Theorem 4 Inequality 119871 (120572minus2)6(119886 119887) lt 119868120572(119886 119887)119871
1minus120572(119886 119887) lt
1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0 with 119886 = 119887and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possible lower andupper Lehmer mean bounds for the product 119868120572(119886 119887)1198711minus120572(119886 119887)
Proof Inequality 119868120572(119886 119887)119871
1minus120572(119886 119887) lt 1198710(119886 119887) follows di-
rectly from (10) and (11)For the other inequality we note that
119868120572(119886 119887) 119871
1minus120572(119886 119887) minus 119871 (120572minus2)6 (119886 119887)
= 119868120572(119886 119887) 119871
1minus120572(119886 119887) minus [11987223 (119886 119887)]
120572[119871minus13 (119886 119887)]
1minus120572
+ [11987223 (119886 119887)]120572[119871minus13 (119886 119887)]
1minus120572minus 119871 (120572minus2)6 (119886 119887)
(57)
Therefore 119868120572(119886 119887)1198711minus120572(119886 119887) gt 119871 (120572minus2)6(119886 119887) follows from(7) and (10) together with Lemma 3 and (57)
Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are thebest possible lower and upper Lehmer mean bounds for theproduct 119868120572(119886 119887)1198711minus120572(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
119868120572(1 119909) 119871
1minus120572(1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
119890120572((log119909)(119909minus1)minus1)
times(1 minus
1
119909
)
1minus120572119909120576
(log119909)1minus120572
] = +infin
119871 (120572minus2)6+120576 (1 1 + 119909) minus 119868120572(1 1 + 119909) 119871
1minus120572(1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
(58)
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Chinese Journal of Mathematics
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 119890120572[(1+119909) log(1+119909)119909minus1]
times
1199091minus120572
[log (1 + 119909)]120572
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus [1 +
120572
2
119909 + 120572(
1
8
120572 minus
1
6
) 1199092+ 119900 (119909
2)]
times [1 + (
1
2
minus
120572
2
) 119909 + (
1
8
1205722minus
1
24
120572 minus
1
12
) 1199092+ 119900 (119909
2)]
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092
minus [1 +
1
2
119909 + (
120572
24
minus
1
12
) 1199092] + 119900 (119909
2)
=
120576
4
1199092+ 119900 (119909
2)
(59)
Equations (58) and (59) imply that for any 0 lt 120576 lt 1there exist 1198831 = 1198831(120572 120576) gt 1 and 1205751 = 1205751(120572 120576) gt 0such that 119868
120572(1 119909)119871
1minus120572(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198831 +infin)
and 119871 (120572minus2)6+120576(1 1 + 119909) gt 119868120572(1 1 + 119909)119871
1minus120572(1 1 + 119909) for
119909 isin (0 1205751)
Theorem 5 Inequality 119871 (120572minus2)6(119886 119887) lt 120572119868(119886 119887) + (1 minus
120572)119871(119886 119887) lt 1198710(119886 119887) holds for any 120572 isin (0 1) and all 119886 119887 gt 0
with 119886 = 119887 and 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best possiblelower and upper Lehmer mean bounds for the sum 120572119868(119886 119887) +
(1 minus 120572)119871(119886 119887)
Proof Inequality 120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) lt 1198710(119886 119887) followsdirectly from (10) and (11) and inequality 119871 (120572minus2)6(119886 119887) lt
120572119868(119886 119887) + (1 minus 120572)119871(119886 119887) follows fromTheorem 4Next we prove that 119871 (120572minus2)6(119886 119887) and 1198710(119886 119887) are the best
possible lower and upper Lehmer mean bounds for the sum120572119868(119886 119887) + (1 minus 120572)119871(119886 119887)
For any 0 lt 120576 lt 1 and 119909 gt 0 from (1) and (2) we have
lim119909rarr+infin
120572119868 (1 119909) + (1 minus 120572) 119871 (1 119909)
119871minus120576 (1 119909)
= lim119909rarr+infin
[
119909minus120576
+ 1
119909120576minus1
+ 1
times(120572119890(log119909)(119909minus1)
119909120576+
(1 minus 120572) (1 minus 1119909)
log119909
119909120576)]
= +infin
(60)
119871 (120572minus2)6+120576 (1 1 + 119909) minus 120572119868 (1 1 + 119909) minus (1 minus 120572) 119871 (1 1 + 119909)
=
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
(61)
Letting 119909 rarr 0 and making use of Taylor expansion one has
(1 + 119909)(120572+4)6+120576
+ 1
(1 + 119909)(120572minus2)6+120576
+ 1
minus 120572119890(1+119909) log(1+119909)119909minus1
minus
(1 minus 120572) 119909
log (1 + 119909)
= 1 +
1
2
119909 + (
120572
24
minus
1
12
+
120576
4
) 1199092+ 119900 (119909
2)
minus 120572 [1 +
1
2
119909 minus
1
24
1199092+ 119900 (119909
2)]
minus (1 minus 120572) [1 +
1
2
119909 minus
1
12
1199092+ 119900 (119909
2)]
=
120576
4
1199092+ 119900 (119909
2)
(62)
Equations (60)-(62) imply that for any 0 lt 120576 lt 1 thereexist 1198832 = 1198832(120572 120576) gt 1 and 1205752 = 1205752(120572 120576) gt 0 such that120572119868(1 119909) + (1 minus 120572)119871(1 119909) gt 119871minus120576(1 119909) for 119909 isin (1198832 +infin) and119871 (120572minus2)6+120576(1 1 + 119909) gt 120572119868(1 1 + 119909) + (1 minus 120572)119871(1 1 + 119909) for119909 isin (0 1205752)
Acknowledgments
This research was supported by the Natural Science Foun-dation of China under Grants 11071069 and 11171307 andthe Natural Science Foundation of Zhejiang Province underGrants LY13H070004 and LY13A010004
References
[1] G Allasia C Giordano and J Pecaric ldquoOn the arithmetic andlogarithmic means with applications to Stirlingrsquos formulardquo Attidel SeminarioMatematico e Fisico dellrsquoUniversita diModena vol47 no 2 pp 441ndash445 1999
[2] H Alzer ldquoUngleichungen fur (119890119886)119886(119887119890)119887rdquo Elemente der Math-ematik vol 40 pp 120ndash123 1985
[3] H Alzer ldquoUngleichungen fur Mittelwerterdquo Archiv der Mathe-matik vol 47 no 5 pp 422ndash426 1986
[4] F Burk ldquoThe geometric logarithmic and arithmetic meaninequalityrdquoThe American Mathematical Monthly vol 94 no 6pp 527ndash528 1987
[5] B C Carlson ldquoThe logarithmic meanrdquo The American Mathe-matical Monthly vol 79 no 6 pp 615ndash618 1972
[6] T P Lin ldquoThe power mean and the logarithmic mesnrdquo TheAmerican Mathematical Monthly vol 81 no 8 pp 879ndash8831974
[7] J Maloney J Heidel and J Pecaric ldquoA reverse Holder typeinequality for the logarithmic mean and generalizationsrdquo Jour-nal of the Australian Mathematical Society Series B-AppliedMathematics vol 41 no 3 pp 401ndash409 2000
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 7
[8] A O Pittenger ldquoInequalities between arithmetic and logarith-mic meansrdquo Univerzitet u Beogradu Publikacije Elektrotehnivckog Fakulteta Serija Matematika vol 678ndash715 pp 15ndash18 1980
[9] AO Pittenger ldquoThe symmetric logarithmic andpowermeansrdquoUniverzitet u Beogradu Publikacije Elektrotehniv ckog FakultetaSerija Matematika vol 678ndash715 pp 19ndash23 1980
[10] J Sandor ldquoInequalities for meansrdquo in Proceedings of the 3rdSymposium of Mathematics and its Applications (Timisoara1989) pp 87ndash90 Romanian Academy Timisoara Romania1990
[11] J Sandor ldquoOn the identric and logarithmicmeansrdquoAequationesMathematicae vol 40 no 1 pp 261ndash270 1990
[12] J Sandor ldquoA note on some inequalities for meansrdquo Archiv derMathematik vol 56 no 5 pp 471ndash473 1991
[13] J Sandor ldquoOn certain identities for meansrdquo Studia UniversitatisBabes-Bolyai Mathematica vol 38 no 4 pp 7ndash14 1993
[14] J Sandor ldquoOn certain inequalities for meansrdquo Journal ofMathematical Analysis and Applications vol 189 no 2 pp 602ndash606 1995
[15] J Sandor ldquoOn refinements of certain inequalities for meansrdquoArchiv der Mathematik vol 31 no 4 pp 279ndash282 1995
[16] J Sandor ldquoTwo inequalities for meansrdquo International Journal ofMathematics and Mathematical Sciences vol 18 no 3 pp 621ndash623 1995
[17] J Sandor ldquoOn certain inequalities for means IIrdquo Journal ofMathematical Analysis and Applications vol 199 no 2 pp 629ndash635 1996
[18] J Sandor ldquoOn certain inequalities for means IIIrdquo Archiv derMathematik vol 76 no 1 pp 34ndash40 2001
[19] H Alzer and S-L Qiu ldquoInequalites for means in two arivari-ablesrdquo Archiv der Mathematik vol 80 no 2 pp 201ndash215 2003
[20] M K Vamanamurthy and M Vuorinen ldquoInequalities formeansrdquo Journal of Mathematical Analysis and Applications vol183 no 1 pp 155ndash166 1994
[21] J Sandor and I Rasa ldquoInequalities for certain means in twoargumentsrdquoNieuwArchief voorWiskunde vol 15 no 41 pp 51ndash55 1997
[22] P Kahlig and J Matkowski ldquoFunctional equations involvingthe logarithmic meanrdquo ZAMM Zeitschrift fur AngewandteMathematik und Mechanik vol 76 no 7 pp 385ndash390 1996
[23] A O Pittenger ldquoThe logarithmic mean in n variablesrdquo Ameri-can Mathematical Monthly vol 92 no 2 pp 99ndash104 1985
[24] G Polya and G Szego Isoperimetric Inequalities in Mathemat-ical Physics Princeton University Press Princeton NJ USA1951
[25] B C Carlson ldquoAlgorithms involving aritmetic and geometricmeansrdquo American Mathematical Monthly vol 78 pp 496ndash5051971
[26] B C Carlson and J L Gustafson ldquoTotal positivity of meanvalues and hypergeometric functionsrdquo SIAM Journal on Math-ematical Analysis vol 14 no 2 pp 389ndash395 1983
[27] P S Bullen D S Mitrinovic and P M VasicMeans and TheirInequalities D Reidel Dordrecht The Netherlands 1988
[28] K B Stolarsky ldquoHoldermeans Lehmermeans and119909minus1 log cosh
xrdquo Journal of Mathematical Analysis and Applications vol 202no 3 pp 810ndash818 1996
[29] Z Liu ldquoRemark on inequalities between Holder and Lehmermeansrdquo Journal of Mathematical Analysis and Applications vol247 no 1 pp 309ndash313 2000
[30] E F Beckenbach ldquoA class of mean value functionsrdquo AmericanMathematical Monthly vol 57 pp 1ndash6 1950
[31] E F Beckenbach and R Bellman Inequalities Springer NewYork NY USA 1965
[32] Z Pales ldquoInequalites for sums of powersrdquo Journal of Mathemat-ical Analysis and Applications vol 131 no 1 pp 265ndash270 1988
[33] K B Stolarsky ldquoThepower and generalized logarithmicmeansrdquoAmerican Mathematical Monthly vol 87 no 7 pp 545ndash5481980
[34] H Alzer ldquoBestmogliche Abschatzungen fur spezielle Mittelw-erterdquo Prirodoslovno-Matematicki Fakultet Sveucilista u Zagrebuvol 23 no 1 pp 331ndash346 1993
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![SWIFT UVOT CALDB RELEASE NOTE - NASA...v sw00158593002uvv_rw.img[] 19764 2.18 0.30 white sw00130088009uwh_rw.img[] 15201 2.31 0.15 Table 3: Fields and sources used to derive wing COGs.](https://static.fdocuments.in/doc/165x107/5f400d31ee38eb6f893d3015/swift-uvot-caldb-release-note-nasa-v-sw00158593002uvvrwimg-19764-218.jpg)
