MohammedAliAlamriandMaslinaDarusdownloads.hindawi.com/journals/tswj/2016/6360250.pdfe Scientic World...
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Research ArticleOn a New Class of 119901-Valent Meromorphic FunctionsDefined in Conic Domains
Mohammed Ali Alamri and Maslina Darus
School ofMathematical Sciences Faculty of Science and Technology Universiti KebangsaanMalaysia 43600 Bangi SelangorMalaysia
Correspondence should be addressed to Maslina Darus maslinaukmedumy
Received 26 April 2016 Accepted 27 June 2016
Academic Editor Jin-Lin Liu
Copyright copy 2016 M A Alamri and M Darus This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We define a new class of multivalent meromorphic functions using the generalised hypergeometric function We derived this classrelated to conic domain It is also shown that this new class of functions under certain conditions becomes a class of starlikefunctions Some results on inclusion and closure properties are also derived
1 Introduction
Let119872119901 denote the class of functions of the form
119891 (119911) =1
119911119901+
infin
sum
119899=1
119886119899119911119899minus119901 119901 isin N = 1 2 3 (1)
which are analytic and 119901-valent in the punctured unit disccentred at origin 119864lowast = 119911 0 lt |119911| lt 1 = 119864 0Also by 119891(119911) ≺ 119892(119911) we mean 119891(119911) is subordinate to 119892(119911)which implies the existence of an analytic function calledSchwartz function 119908(119911) with |119908(119911)| lt 1 for 119911 isin 119864lowast such that119891(119911) = 119892(119908(119911)) where 119891(119911) and 119892(119911) are multivalent
meromorphic functions Note that if 119892 is univalent in 119864 thenthe above subordination is equivalent to 119891(0) = 119892(0) and119891(119864) sub 119892(119864)
The set of points for 0 lt 120574 lt 1 and 119896 isin [0infin)
Ω119896120574 = 120574Ω119896 + (1 minus 120574) (2)
where
Ω119896 = 119906 + 119894V 119906 gt 119896radic(119906 minus 1)2+ V2 119906 gt 0 (3)
[1] showed that the extremal functions 119902119896120574(119911) for conicregions are convex univalent and given by
119902119896120574 (119911) =
1 + (1 minus 2120574) 119911
1 minus 119911 119896 = 0
1 +2120574
1 minus 1198962[2
120587(arccos 119896) arctanhradic119911] 0 lt 119896 lt 1
1 +2120574
1205872(log 1 +
radic119911
1 minus radic119911)
2
119896 = 1
1 +120574
1198962 minus 1
[[
[
sin( 120587
2119877 (119905)int
119906(119911)radic119905
0
1
radic1 minus 1199092radic1 minus (119905119909)2
d119909) minus 1]]
]
119896 gt 1
(4)
where 119877(119905) is Legendrersquos complete elliptic integral of the firstkind with 1198771015840(119905) = radic1 minus 1199052 as its complementary integral
119906(119911) = (119911 minus radic119905)(1 minus radic119905119911) 119905 isin (0 1) and 119911 isin 119864 is chosenin such a way that 119896 = cosh(1205871198771015840(119905)119877(119905))
Hindawi Publishing Corporatione Scientific World JournalVolume 2016 Article ID 6360250 7 pageshttpdxdoiorg10115520166360250
2 The Scientific World Journal
The generalised hypergeometric function119902119865119904(1205721 120572119902
1205731 120573119904 119911) for complex parameters1205721 120572119902 and1205731 120573119904with 120573119895 = 0 minus1 minus2 minus3 for 119895 = 1 2 3 119904 is defined as
119902119865119904 (1205721 120572119902 1205731 120573119904 119911) =
infin
sum
119899=0
(1205721)119899sdot sdot sdot (120572119902)119899
(1205731)119899sdot sdot sdot (120573119904)119899
119899119911119899
= 1 +
infin
sum
119899=1
(1205721)119899sdot sdot sdot (120572119902)119899
(1205731)119899sdot sdot sdot (120573119904)119899
119911119899
(119899)
(5)
with 119902 le 119904+1 119902 119904 isin N0 = Ncup0 and (120572)119899 is the well-knownPochhammer symbol related to the factorial and the Gammafunction by the relation
(120572)119899 =(120572 + 119899 minus 1)
(120572 minus 1)=Γ (120572 + 119899)
Γ (120572) (6)
Also (5) implies
ℎ119901 (1205721 120572119902 1205731 120573119904 119911)
= 119911minus119901
119902119865119904 (1205721 120572119902 1205731 120573119904 119911) isin 119872119901
(7)
Liu and Srivastava [2] defined a linear operator for func-tions belonging to the class of multivalent meromorphicfunction119867119901(1205721 120572119902 1205731 120573119904) 119872119901 rarr119872119901 as follows
119867119901 (1205721 120572119902 1205731 120573119904)
= ℎ119901 (1205721 120572119902 1205731 120573119904 119911) ⋆ 119891 (119911)
(8)
If we assume for brevity that 119867119901119902119904(1205721) = 119867119901(1205721 120572119902
1205731 120573119904) then the following identity holds for this operator
119911 [119867119901119902119904 (1205721) 119891 (119911)]1015840
= 1205721119867119901119902119904 (1205721 + 1) 119891 (119911)
minus (1205721 + 119901)119867119901119902119904 (1205721) 119891 (119911)
(9)
Shareef [3] defined and studied subclass119872119876119901(119896 120582 1205721) ofmeromorphic function associated with conic domain for 119896 ge0 0 le 120582 lt 1 and 119901 ge 1 as follows
minus1
119901
[
[
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840]
]
≺ 119902119896120574 (119911)
(10)
We now define a new subclass119872119876119901(119887 119896 120582 1205721) of mero-morphic function associated with conic domain for 119896 ge 00 le 120582 lt 1 119901 ge 1 and 119887 ge 1 as follows
minus1
119901
[
[
1
+1
119887(
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)
minus 119887]
]
≺ 119902119896120574 (119911)
(11)
Since 119902119896120574 is a convex and univalent function for ℎ(119911) ≺119902119896120574(119911) it means ℎ(119864lowast) is contained in 119902119896120574(119864
lowast) where
ℎ (119911) = minus1
119901
[
[
1
+1
119887(
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)
minus 119887]
]
(12)
In the next two sections for brevity we drop the subscripts ofthe operator119867119901119902119904(1205721)
2 Preliminary Results
Lemma 1 (see [4]) Let ℎ2(119911) be convex in 119864 andR(120582ℎ2(119911) +120583) gt 0 where 120583 isin C 120582 isin C0 and 119911 isin 119864 If ℎ1(119911) is analyticin 119864 with ℎ1(0) = ℎ2(0) then
ℎ1 (119911) +119911ℎ1015840
1(119911)
120582ℎ1 (119911) + 120583≺ ℎ2 (119911)
119894119898119901119897119894119890119904 ℎ1 (119911) ≺ ℎ2 (119911)
(13)
Lemma 2 (see [5]) Let ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 and 119867(119911) =
1 + suminfin
119899=1119889119899119911119899 and ℎ ≺ 119867 If 119867(119911) is univalent and convex in
119864 then10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611988911003816100381610038161003816 (14)
for 119899 ge 1
Lemma 3 (see [6]) If 119902119896120574(119911) = 1 + 1199021119911 + 11990221199112 + sdot sdot sdot then
1199021 =
8 (1 minus 120574) (arccos (119896))2
1205872 (1 minus 1198962) 0 lt 119896 lt 1
8 (1 minus 120574)
1205872 119896 = 1
1205872(1 minus 120574)
4radic119905 (1198962 minus 1) 1198962 (119905) (1 + 119905) 119896 gt 1
(15)
One now states and proves the main results
3 Main Results
In this section we explore some of the geometric propertiesexhibited by the class119872119876119901(119887 119896 120582 1205721)
We begin by discussing an inclusion property for the class119872119876119901(119887 119896 120582 1205721)
Theorem 4 IfR(1205721) gt 119901R(119887119902119896120582(119911) minus 1) then
119872119876119901 (119887 119896 120582 1205721 + 1) sub 119872119876119901 (119887 119896 120582 1205721) (16)
The Scientific World Journal 3
Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)
minus 119887] = ℎ (119911)
(17)
But differentiating (9) with respect to 119911 we get
[119911119867 (1205721) 119891 (119911)]1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]
1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
1199112(119867 (1205721) 119891 (119911))
10158401015840
= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840
(18)
Putting (18) in (17) we have
1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))
1015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911)
1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911) + (1205721 + 119901)
(19)
Taking logarithmic derivative of (19) we have
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ
1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)
minus1
119901[1
+1
119887(119911 (119867 (1205721 + 1) 119891 (119911))
1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)
minus 119887] = ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)
(20)
Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore
ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)
Using Lemma 2 we have
ℎ (119911) ≺ 119902119896120574 (119911) (22)
providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)
We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral
Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral
119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901
119911120578int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)
maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)
Proof From (23) we have
119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)
Note that
119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)
Differentiating (24) above we get
120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))
1015840
= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))
4 The Scientific World Journal
120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))
(26)
Differentiate again
120578 (119861120578 (119867 (120572) 119891 (119911)))1015840
+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840
+ (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840
119911 (119861120578 (119891 (119911)))10158401015840
= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840
minus (120578 + 1) (119861120578 (119891 (119911)))1015840
(27)
Now let
minus1
119901
[
[
1
+1
119887(
119911 (119861120578 (119867 (1205721) 119891))1015840
+ 1205821199112(119861120578 (119867 (1205721) 119891))
10158401015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)
minus 119887]
]
= 119892 (119911)
(28)
Using (26) and (27) in (28) we get
119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)
1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))
1015840
]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))
1015840
minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887
2minus 119887 + 120578
(29)
Now taking logarithmic derivative we have
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)
1015840+ (1 minus 120582) (119867 (1205721) 119891)
1015840]
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
minus
(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))
10158401015840
+ 120582 (119861120578 (119867 (1205721) 119891))1015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
=minus1198871199011198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
minus1
119901[1 +
1
119887(119911 (119867 (1205721) 119891)
1015840+ 1205821199112(119867 (1205721) 119891)
10158401015840
(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +
1199111198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
(30)
Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies
119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)
This proves the assertion
Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)
Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)
provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
The Scientific World Journal 5
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(33)
for all 119896 le 119899
Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= 119887ℎ (119911) +119887
119901minus1198872
119901
(34)
which gives
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901
= [119887ℎ (119911) +119887 (1 minus 119887)
119901]
sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]
(35)
Let us write119867(1205721)119891(119911) = 119891(119911) then
119891 (119911) = 119911minus119901+
infin
sum
119899=0
119886119899119911119899minus119901
1198911015840(119911) = minus119901119911
minus119901minus1+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901minus1
1199111198911015840(119911) = minus119901119911
minus119901+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901
11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901minus2
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2
119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901
(36)
Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes
119911minus119901minus
infin
sum
119899=0
119886119899
(119899 minus 119901)
119901119911119899minus119901+ 120582(minus (119901 + 1) 119911
minus119901
+
infin
sum
119899=0
119886119899
(119899 minus 119901) (119899 minus 119901 minus 1)
119901119911119899minus119901)
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887
+
infin
sum
119899=1
119887119888119899119911119899+119887 (1 minus 119887)
119901] [(1 minus 120582119901 minus 120582) 119911
minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(37)
From (37) we have
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887 + 1198871198881119911
+ 11988711988821199112+ 1198871198883119911
3+ sdot sdot sdot + 119887119888119899119911
119899+ sdot sdot sdot +
119887 (1 minus 119887)
119901]
sdot [(1 minus 120582119901 minus 120582) 119911minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(38)
Now comparing coefficients of 1199111minus119901 we have
minus 1198861
1 minus 119901
119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (
119887 (1 minus 119887)
119901)
sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)
minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1
119901]
= 1198871198881 (1 minus 120582119901 minus 120582)
1198861 = minus119901
(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)
sdot 1198871198881
(39)
6 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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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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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
2 The Scientific World Journal
The generalised hypergeometric function119902119865119904(1205721 120572119902
1205731 120573119904 119911) for complex parameters1205721 120572119902 and1205731 120573119904with 120573119895 = 0 minus1 minus2 minus3 for 119895 = 1 2 3 119904 is defined as
119902119865119904 (1205721 120572119902 1205731 120573119904 119911) =
infin
sum
119899=0
(1205721)119899sdot sdot sdot (120572119902)119899
(1205731)119899sdot sdot sdot (120573119904)119899
119899119911119899
= 1 +
infin
sum
119899=1
(1205721)119899sdot sdot sdot (120572119902)119899
(1205731)119899sdot sdot sdot (120573119904)119899
119911119899
(119899)
(5)
with 119902 le 119904+1 119902 119904 isin N0 = Ncup0 and (120572)119899 is the well-knownPochhammer symbol related to the factorial and the Gammafunction by the relation
(120572)119899 =(120572 + 119899 minus 1)
(120572 minus 1)=Γ (120572 + 119899)
Γ (120572) (6)
Also (5) implies
ℎ119901 (1205721 120572119902 1205731 120573119904 119911)
= 119911minus119901
119902119865119904 (1205721 120572119902 1205731 120573119904 119911) isin 119872119901
(7)
Liu and Srivastava [2] defined a linear operator for func-tions belonging to the class of multivalent meromorphicfunction119867119901(1205721 120572119902 1205731 120573119904) 119872119901 rarr119872119901 as follows
119867119901 (1205721 120572119902 1205731 120573119904)
= ℎ119901 (1205721 120572119902 1205731 120573119904 119911) ⋆ 119891 (119911)
(8)
If we assume for brevity that 119867119901119902119904(1205721) = 119867119901(1205721 120572119902
1205731 120573119904) then the following identity holds for this operator
119911 [119867119901119902119904 (1205721) 119891 (119911)]1015840
= 1205721119867119901119902119904 (1205721 + 1) 119891 (119911)
minus (1205721 + 119901)119867119901119902119904 (1205721) 119891 (119911)
(9)
Shareef [3] defined and studied subclass119872119876119901(119896 120582 1205721) ofmeromorphic function associated with conic domain for 119896 ge0 0 le 120582 lt 1 and 119901 ge 1 as follows
minus1
119901
[
[
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840]
]
≺ 119902119896120574 (119911)
(10)
We now define a new subclass119872119876119901(119887 119896 120582 1205721) of mero-morphic function associated with conic domain for 119896 ge 00 le 120582 lt 1 119901 ge 1 and 119887 ge 1 as follows
minus1
119901
[
[
1
+1
119887(
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)
minus 119887]
]
≺ 119902119896120574 (119911)
(11)
Since 119902119896120574 is a convex and univalent function for ℎ(119911) ≺119902119896120574(119911) it means ℎ(119864lowast) is contained in 119902119896120574(119864
lowast) where
ℎ (119911) = minus1
119901
[
[
1
+1
119887(
119911 (119867119901119902119904 (1205721) 119891 (119911))1015840
+ 1205821199112(119867119901119902119904 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867119901119902119904 (1205721) 119891 (119911) + 120582119911 (119867119901119902119904 (1205721) 119891 (119911))1015840)
minus 119887]
]
(12)
In the next two sections for brevity we drop the subscripts ofthe operator119867119901119902119904(1205721)
2 Preliminary Results
Lemma 1 (see [4]) Let ℎ2(119911) be convex in 119864 andR(120582ℎ2(119911) +120583) gt 0 where 120583 isin C 120582 isin C0 and 119911 isin 119864 If ℎ1(119911) is analyticin 119864 with ℎ1(0) = ℎ2(0) then
ℎ1 (119911) +119911ℎ1015840
1(119911)
120582ℎ1 (119911) + 120583≺ ℎ2 (119911)
119894119898119901119897119894119890119904 ℎ1 (119911) ≺ ℎ2 (119911)
(13)
Lemma 2 (see [5]) Let ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 and 119867(119911) =
1 + suminfin
119899=1119889119899119911119899 and ℎ ≺ 119867 If 119867(119911) is univalent and convex in
119864 then10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611988911003816100381610038161003816 (14)
for 119899 ge 1
Lemma 3 (see [6]) If 119902119896120574(119911) = 1 + 1199021119911 + 11990221199112 + sdot sdot sdot then
1199021 =
8 (1 minus 120574) (arccos (119896))2
1205872 (1 minus 1198962) 0 lt 119896 lt 1
8 (1 minus 120574)
1205872 119896 = 1
1205872(1 minus 120574)
4radic119905 (1198962 minus 1) 1198962 (119905) (1 + 119905) 119896 gt 1
(15)
One now states and proves the main results
3 Main Results
In this section we explore some of the geometric propertiesexhibited by the class119872119876119901(119887 119896 120582 1205721)
We begin by discussing an inclusion property for the class119872119876119901(119887 119896 120582 1205721)
Theorem 4 IfR(1205721) gt 119901R(119887119902119896120582(119911) minus 1) then
119872119876119901 (119887 119896 120582 1205721 + 1) sub 119872119876119901 (119887 119896 120582 1205721) (16)
The Scientific World Journal 3
Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)
minus 119887] = ℎ (119911)
(17)
But differentiating (9) with respect to 119911 we get
[119911119867 (1205721) 119891 (119911)]1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]
1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
1199112(119867 (1205721) 119891 (119911))
10158401015840
= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840
(18)
Putting (18) in (17) we have
1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))
1015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911)
1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911) + (1205721 + 119901)
(19)
Taking logarithmic derivative of (19) we have
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ
1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)
minus1
119901[1
+1
119887(119911 (119867 (1205721 + 1) 119891 (119911))
1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)
minus 119887] = ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)
(20)
Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore
ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)
Using Lemma 2 we have
ℎ (119911) ≺ 119902119896120574 (119911) (22)
providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)
We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral
Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral
119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901
119911120578int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)
maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)
Proof From (23) we have
119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)
Note that
119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)
Differentiating (24) above we get
120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))
1015840
= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))
4 The Scientific World Journal
120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))
(26)
Differentiate again
120578 (119861120578 (119867 (120572) 119891 (119911)))1015840
+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840
+ (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840
119911 (119861120578 (119891 (119911)))10158401015840
= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840
minus (120578 + 1) (119861120578 (119891 (119911)))1015840
(27)
Now let
minus1
119901
[
[
1
+1
119887(
119911 (119861120578 (119867 (1205721) 119891))1015840
+ 1205821199112(119861120578 (119867 (1205721) 119891))
10158401015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)
minus 119887]
]
= 119892 (119911)
(28)
Using (26) and (27) in (28) we get
119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)
1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))
1015840
]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))
1015840
minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887
2minus 119887 + 120578
(29)
Now taking logarithmic derivative we have
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)
1015840+ (1 minus 120582) (119867 (1205721) 119891)
1015840]
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
minus
(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))
10158401015840
+ 120582 (119861120578 (119867 (1205721) 119891))1015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
=minus1198871199011198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
minus1
119901[1 +
1
119887(119911 (119867 (1205721) 119891)
1015840+ 1205821199112(119867 (1205721) 119891)
10158401015840
(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +
1199111198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
(30)
Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies
119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)
This proves the assertion
Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)
Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)
provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
The Scientific World Journal 5
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(33)
for all 119896 le 119899
Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= 119887ℎ (119911) +119887
119901minus1198872
119901
(34)
which gives
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901
= [119887ℎ (119911) +119887 (1 minus 119887)
119901]
sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]
(35)
Let us write119867(1205721)119891(119911) = 119891(119911) then
119891 (119911) = 119911minus119901+
infin
sum
119899=0
119886119899119911119899minus119901
1198911015840(119911) = minus119901119911
minus119901minus1+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901minus1
1199111198911015840(119911) = minus119901119911
minus119901+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901
11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901minus2
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2
119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901
(36)
Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes
119911minus119901minus
infin
sum
119899=0
119886119899
(119899 minus 119901)
119901119911119899minus119901+ 120582(minus (119901 + 1) 119911
minus119901
+
infin
sum
119899=0
119886119899
(119899 minus 119901) (119899 minus 119901 minus 1)
119901119911119899minus119901)
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887
+
infin
sum
119899=1
119887119888119899119911119899+119887 (1 minus 119887)
119901] [(1 minus 120582119901 minus 120582) 119911
minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(37)
From (37) we have
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887 + 1198871198881119911
+ 11988711988821199112+ 1198871198883119911
3+ sdot sdot sdot + 119887119888119899119911
119899+ sdot sdot sdot +
119887 (1 minus 119887)
119901]
sdot [(1 minus 120582119901 minus 120582) 119911minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(38)
Now comparing coefficients of 1199111minus119901 we have
minus 1198861
1 minus 119901
119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (
119887 (1 minus 119887)
119901)
sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)
minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1
119901]
= 1198871198881 (1 minus 120582119901 minus 120582)
1198861 = minus119901
(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)
sdot 1198871198881
(39)
6 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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The Scientific World Journal 3
Proof Let 119891 isin 119872119876119901(119887 119896 120582 1205721 + 1) and set
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840)
minus 119887] = ℎ (119911)
(17)
But differentiating (9) with respect to 119911 we get
[119911119867 (1205721) 119891 (119911)]1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
119911 [119867 (1205721) 119891 (119911)]10158401015840+ [119867 (1205721) 119891 (119911)]
1015840
= 1205721119867((1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901) (119867 (1205721) 119891 (119911))1015840
1199112(119867 (1205721) 119891 (119911))
10158401015840
= 1205721119911 (119867 (1205721 + 1) 119891 (119911))1015840
minus (1205721 + 119901 + 1) 119911 (119867 (1205721) 119891 (119911))1015840
(18)
Putting (18) in (17) we have
1205821199111205721 (119867 (120572 + 1) 119891 (119911))1015840+ [1 minus 120582 (1205721 + 119901 + 1)] 119911 (119867 (1205721) 119891 (119911))
1015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911)
1205821205721119911 (119867 (1205721 + 1) 119891 (119911))1015840+ (1 minus 120582) 1205721 (119867 (1205721 + 1) 119891 (119911))
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= minus119887119901ℎ (119911) + (1205721 + 119901)
(19)
Taking logarithmic derivative of (19) we have
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
+ 119887119901ℎ (119911) + 119887 (119887 minus 1) =minus119887119901119911ℎ
1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901
119911 (119867 (1205721 + 1) 119891 (119911))1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582) (119867 (1205721 + 1) 119891 (119911)) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840
= minus119887119901[ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + 1205721 + 119901] minus 119887 (119887 minus 1)
minus1
119901[1
+1
119887(119911 (119867 (1205721 + 1) 119891 (119911))
1015840+ 1205821199112(119867 (1205721 + 1) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721 + 1) 119891 (119911) + 120582119911 (119867 (1205721 + 1) 119891 (119911))1015840)
minus 119887] = ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)
(20)
Since 119891 isin 119872119876119902(119887 119896 120582 1205721 + 1) therefore
ℎ (119911) +119911ℎ1015840(119911)
minus119887119901ℎ (119911) + (1205721 + 119901)≺ 119902119896120574 (119911) (21)
Using Lemma 2 we have
ℎ (119911) ≺ 119902119896120574 (119911) (22)
providedR(minus119887119901119902119896120582(119911)+1205721 +119901) gt 0 or equivalentlyR(1205721) gt119901R(119887119902119896120574(119911) minus 1) Hence 119891 isin 119872119876119901(119887 119896 120582 1205721)
We now show that the class 119872119876119901(119887 119896 120582 1205721) is closedunder a certain integral
Theorem 5 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then the integral
119861120578 (119867 (120572) 119891 (119911)) =120578 minus 119901
119911120578int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (23)
maps 119891(119911) into119872119876119901(119887 119896 120582 1205721)
Proof From (23) we have
119911120578119861120578 (119867 (120572) 119891 (119911)) = (120578 minus 119901)int
119911
0
119905120578minus1(119867 (120572) 119891 (119911)) d119905 (24)
Note that
119861120578 (119867 (120572) 119891 (119911)) = (119867 (120572)) 119861120578119891 (119911) (25)
Differentiating (24) above we get
120578119911120578minus1119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))
1015840
= (120578 minus 119901) 119911120578minus1(119867 (120572) 119891 (119911))
4 The Scientific World Journal
120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))
(26)
Differentiate again
120578 (119861120578 (119867 (120572) 119891 (119911)))1015840
+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840
+ (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840
119911 (119861120578 (119891 (119911)))10158401015840
= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840
minus (120578 + 1) (119861120578 (119891 (119911)))1015840
(27)
Now let
minus1
119901
[
[
1
+1
119887(
119911 (119861120578 (119867 (1205721) 119891))1015840
+ 1205821199112(119861120578 (119867 (1205721) 119891))
10158401015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)
minus 119887]
]
= 119892 (119911)
(28)
Using (26) and (27) in (28) we get
119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)
1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))
1015840
]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))
1015840
minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887
2minus 119887 + 120578
(29)
Now taking logarithmic derivative we have
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)
1015840+ (1 minus 120582) (119867 (1205721) 119891)
1015840]
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
minus
(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))
10158401015840
+ 120582 (119861120578 (119867 (1205721) 119891))1015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
=minus1198871199011198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
minus1
119901[1 +
1
119887(119911 (119867 (1205721) 119891)
1015840+ 1205821199112(119867 (1205721) 119891)
10158401015840
(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +
1199111198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
(30)
Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies
119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)
This proves the assertion
Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)
Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)
provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
The Scientific World Journal 5
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(33)
for all 119896 le 119899
Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= 119887ℎ (119911) +119887
119901minus1198872
119901
(34)
which gives
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901
= [119887ℎ (119911) +119887 (1 minus 119887)
119901]
sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]
(35)
Let us write119867(1205721)119891(119911) = 119891(119911) then
119891 (119911) = 119911minus119901+
infin
sum
119899=0
119886119899119911119899minus119901
1198911015840(119911) = minus119901119911
minus119901minus1+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901minus1
1199111198911015840(119911) = minus119901119911
minus119901+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901
11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901minus2
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2
119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901
(36)
Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes
119911minus119901minus
infin
sum
119899=0
119886119899
(119899 minus 119901)
119901119911119899minus119901+ 120582(minus (119901 + 1) 119911
minus119901
+
infin
sum
119899=0
119886119899
(119899 minus 119901) (119899 minus 119901 minus 1)
119901119911119899minus119901)
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887
+
infin
sum
119899=1
119887119888119899119911119899+119887 (1 minus 119887)
119901] [(1 minus 120582119901 minus 120582) 119911
minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(37)
From (37) we have
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887 + 1198871198881119911
+ 11988711988821199112+ 1198871198883119911
3+ sdot sdot sdot + 119887119888119899119911
119899+ sdot sdot sdot +
119887 (1 minus 119887)
119901]
sdot [(1 minus 120582119901 minus 120582) 119911minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(38)
Now comparing coefficients of 1199111minus119901 we have
minus 1198861
1 minus 119901
119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (
119887 (1 minus 119887)
119901)
sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)
minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1
119901]
= 1198871198881 (1 minus 120582119901 minus 120582)
1198861 = minus119901
(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)
sdot 1198871198881
(39)
6 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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4 The Scientific World Journal
120578119861120578 (119867 (120572) 119891 (119911)) + 119911 (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))
(26)
Differentiate again
120578 (119861120578 (119867 (120572) 119891 (119911)))1015840
+ 119911 (119861120578 (119867 (120572) 119891 (119911)))10158401015840
+ (119861120578 (119867 (120572) 119891 (119911)))1015840
= (120578 minus 119901) (119867 (120572) 119891 (119911))1015840
119911 (119861120578 (119891 (119911)))10158401015840
= (120578 minus 119901) (119861120578 (119867 (120572) 119891 (119911)))1015840
minus (120578 + 1) (119861120578 (119891 (119911)))1015840
(27)
Now let
minus1
119901
[
[
1
+1
119887(
119911 (119861120578 (119867 (1205721) 119891))1015840
+ 1205821199112(119861120578 (119867 (1205721) 119891))
10158401015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840)
minus 119887]
]
= 119892 (119911)
(28)
Using (26) and (27) in (28) we get
119911119861120578 (119867 (1205721) 119891)1015840+ 120582 [119911 (120578 minus 119901) (119867 (1205721) 119891)
1015840minus 119911 (1 + 120578) (119861120578 (119867 (1205721) 119891))
1015840
]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
120582119911 (120578 minus 119901) (119867 (1205721) 119891)1015840+ [1 minus 120582 (1 + 120578)] [(120578 minus 119901) (119867 (1205721) 119891) minus 120578 (119861120578 (119867 (1205721) 119891))]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)] minus 120582120578119911 (119861120578 (119867 (1205721) 119891))
1015840
minus (1 minus 120582) 120578 (119861120578 (119867 (1205721) 119891))
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
= minus119887119901119892 (119911) + 1198872minus 119887
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840= minus119887119901119892 (119911) + 119887
2minus 119887 + 120578
(29)
Now taking logarithmic derivative we have
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)10158401015840+ 120582 (119867 (1205721) 119891)
1015840+ (1 minus 120582) (119867 (1205721) 119891)
1015840]
(120578 minus 119901) [120582119911 (119867 (1205721) 119891)1015840minus (1 minus 120582) (119867 (1205721) 119891)]
minus
(1 minus 120582) 119861120578 (119867 (1205721) 119891)1015840+ 120582119911 (119861120578 (119867 (1205721) 119891))
10158401015840
+ 120582 (119861120578 (119867 (1205721) 119891))1015840
(1 minus 120582) 119861120578 (119867 (1205721) 119891) + 120582119911 (119861120578 (119867 (1205721) 119891))1015840
=minus1198871199011198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
minus1
119901[1 +
1
119887(119911 (119867 (1205721) 119891)
1015840+ 1205821199112(119867 (1205721) 119891)
10158401015840
(1 minus 120582) (119867 (1205721) 119891) + 120582119911 (119867 (1205721) 119891)1015840) minus 119887] = 119892 (119911) +
1199111198921015840(119911)
minus119887119901119892 (119911) + 1198872 minus 119887
(30)
Using Lemma 2 we get 119892(119911) ≺ 119902119896120574(119911) which implies
119861120578 (119891 (119911)) isin 119872119876119901 (119887 119896 120582 1205721) (31)
This proves the assertion
Now we get coefficient estimates of the class 119872119876119901(119887 119896120582 1205721)
Theorem 6 If 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) and 119891(119911) is given by(1) then
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 1198860 = 1 (32)
provided2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
The Scientific World Journal 5
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(33)
for all 119896 le 119899
Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= 119887ℎ (119911) +119887
119901minus1198872
119901
(34)
which gives
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901
= [119887ℎ (119911) +119887 (1 minus 119887)
119901]
sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]
(35)
Let us write119867(1205721)119891(119911) = 119891(119911) then
119891 (119911) = 119911minus119901+
infin
sum
119899=0
119886119899119911119899minus119901
1198911015840(119911) = minus119901119911
minus119901minus1+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901minus1
1199111198911015840(119911) = minus119901119911
minus119901+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901
11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901minus2
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2
119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901
(36)
Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes
119911minus119901minus
infin
sum
119899=0
119886119899
(119899 minus 119901)
119901119911119899minus119901+ 120582(minus (119901 + 1) 119911
minus119901
+
infin
sum
119899=0
119886119899
(119899 minus 119901) (119899 minus 119901 minus 1)
119901119911119899minus119901)
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887
+
infin
sum
119899=1
119887119888119899119911119899+119887 (1 minus 119887)
119901] [(1 minus 120582119901 minus 120582) 119911
minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(37)
From (37) we have
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887 + 1198871198881119911
+ 11988711988821199112+ 1198871198883119911
3+ sdot sdot sdot + 119887119888119899119911
119899+ sdot sdot sdot +
119887 (1 minus 119887)
119901]
sdot [(1 minus 120582119901 minus 120582) 119911minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(38)
Now comparing coefficients of 1199111minus119901 we have
minus 1198861
1 minus 119901
119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (
119887 (1 minus 119887)
119901)
sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)
minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1
119901]
= 1198871198881 (1 minus 120582119901 minus 120582)
1198861 = minus119901
(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)
sdot 1198871198881
(39)
6 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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The Scientific World Journal 5
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582
119891119900119903 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(33)
for all 119896 le 119899
Proof Let 119891(119911) isin 119872119876119901(119887 119896 120582 1205721) then by definition wehave
minus1
119901[1
+1
119887(119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
minus 119887)] = ℎ (119911) ≺ 119902119896120574 (119911)
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901 (1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840
= 119887ℎ (119911) +119887
119901minus1198872
119901
(34)
which gives
minus119911 (119867 (1205721) 119891 (119911))
1015840+ 1205821199112(119867 (1205721) 119891 (119911))
10158401015840
119901
= [119887ℎ (119911) +119887 (1 minus 119887)
119901]
sdot [(1 minus 120582)119867 (1205721) 119891 (119911) + 120582119911 (119867 (1205721) 119891 (119911))1015840]
(35)
Let us write119867(1205721)119891(119911) = 119891(119911) then
119891 (119911) = 119911minus119901+
infin
sum
119899=0
119886119899119911119899minus119901
1198911015840(119911) = minus119901119911
minus119901minus1+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901minus1
1199111198911015840(119911) = minus119901119911
minus119901+
infin
sum
119899=0
119886119899 (119899 minus 119901) 119911119899minus119901
11989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901minus2
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901minus2
119911211989110158401015840(119911) = minus119901 (minus119901 minus 1) 119911
minus119901
+
infin
sum
119899=0
119886119899 (119899 minus 119901) (119899 minus 119901 minus 1) 119911119899minus119901
(36)
Assuming ℎ(119911) = 1 + suminfin119899=1119888119899119911119899 then (35) becomes
119911minus119901minus
infin
sum
119899=0
119886119899
(119899 minus 119901)
119901119911119899minus119901+ 120582(minus (119901 + 1) 119911
minus119901
+
infin
sum
119899=0
119886119899
(119899 minus 119901) (119899 minus 119901 minus 1)
119901119911119899minus119901)
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887
+
infin
sum
119899=1
119887119888119899119911119899+119887 (1 minus 119887)
119901] [(1 minus 120582119901 minus 120582) 119911
minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(37)
From (37) we have
(1 minus 120582119901 minus 120582) 119911minus119901
minus
infin
sum
119899=1
119886119899
119899 minus 119901
119901[1 + 120582 (119899 minus 119901 minus 1)] 119911
119899minus119901= [119887 + 1198871198881119911
+ 11988711988821199112+ 1198871198883119911
3+ sdot sdot sdot + 119887119888119899119911
119899+ sdot sdot sdot +
119887 (1 minus 119887)
119901]
sdot [(1 minus 120582119901 minus 120582) 119911minus119901
+
infin
sum
119899=1
119886119899 [1 + 120582 (119899 minus 119901 minus 1)] 119911119899minus119901]
(38)
Now comparing coefficients of 1199111minus119901 we have
minus 1198861
1 minus 119901
119901(1 minus 120582119901) = 1198861119887 (1 minus 120582119901) + 1198861 (
119887 (1 minus 119887)
119901)
sdot (1 minus 120582119901) + 1198871198881 (1 minus 120582119901 minus 120582)
minus 1198861 (1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1
119901]
= 1198871198881 (1 minus 120582119901 minus 120582)
1198861 = minus119901
(1 minus 120582119901) [119887 (1 minus 119887) + 119887119901 minus 119901 + 1](1 minus 120582119901 minus 120582)
sdot 1198871198881
(39)
6 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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 The Scientific World Journal
and comparing the coefficients of 1199112minus119901 gives
minus 1198862 [2 minus 119901
119901] [1 + 120582 (1 minus 119901)] = 1198862 [1 + 120582 (1 minus 119901)] [119887
+119887 (1 minus 119887)
119901] + 1198861 [1 minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [2 minus 119901
119901+119887119901 + 119887 (1 minus 119887)
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
minus 1198862 [1 + 120582 (1 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 2
119901] = 1198861 [1
minus 120582119901] 1198871198881 + 1198871198882 (1 minus 120582119901 minus 120582)
1198862 = minus119901
(1 + 120582 (1 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 2)((1
minus 120582119901 minus 120582) 1198871198882 + [1 minus 120582119901] 11988711988611198881)
(40)
and for the coefficient of 1199113minus119901 we have
minus 1198863 [1 + 120582 (2 minus 119901)] [119887 (1 minus 119887) + 119887119901 minus 119901 + 3
119901]
= 11988711988811198862 [1 + 120582 (1 minus 119901)] + 11988711988821198861 [1 minus 120582119901] + 1198871198883 [1
minus 120582119901 minus 120582]
1198863 = minus119901
(1 + 120582 (2 minus 119901)) (119887 (1 minus 119887) + 119887119901 minus 119901 + 3)([1
minus 120582119901 minus 120582] 1198871198883 + [1 minus 120582119901] 11988711988821198861 + [1 + 120582 (1 minus 119901)]
sdot 11988711988811198862)
(41)
which generalise to
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)[(1
minus 120582119901 minus 120582) 119887119888119899 + (1 minus 120582119901) 119887119888119899minus11198861 + (1 minus 120582119901 + 120582)
sdot 119887119888119899minus21198862 + sdot sdot sdot + (1 minus 120582119901 + (119899 minus 2) 120582) 1198871198881119886119899minus1]
(42)
The above expression can also be written as
119886119899 = minus119901
(1 minus 120582119901 + (119899 minus 1) 120582) (119887 (1 minus 119887) + 119887119901 minus 119901 + 119899)
sdot
119899
sum
119896=1
[1 minus 120582119901 + (119896 minus 2) 120582] 119888119899minus119896minus1119886119896minus1
with 1198860 = 1
(43)
Now taking
119860119899119896 = minus1 minus 120582119901 + (119896 minus 2) 120582
1 minus 120582119901 + (119899 minus 1) 120582 119899 ge 119896 (44)
we have1003816100381610038161003816119860119899119896
1003816100381610038161003816 lt 1
if
2 (120582119901 minus 1) + (3 minus 119899) 120582 lt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 gt 0
2 (120582119901 minus 1) + (3 minus 119899) 120582 gt 119896120582 for 1 minus 120582119901 + (119899 minus 1) 120582 lt 0
(45)
for all 119899 ge 119896 Since 119902119896120574(119911) is univalent and 119902119896120574(119864) is convexapplying Rogosinskirsquos theorem we have
10038161003816100381610038161198881198991003816100381610038161003816 le100381610038161003816100381611990211003816100381610038161003816 (46)
where 1199021 is given in (15) Under the conditions given in (45)expressions (39)ndash(42) give
100381610038161003816100381611988611003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 1
10038161003816100381610038161003816100381610038161003816
100381610038161003816100381611988621003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 2
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816)
100381610038161003816100381611988631003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 3
10038161003816100381610038161003816100381610038161003816
(1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816)
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
sdot (1 +100381610038161003816100381611988611003816100381610038161003816 +100381610038161003816100381611988621003816100381610038161003816 + sdot sdot sdot +
1003816100381610038161003816119886119899minus11003816100381610038161003816)
(47)
This can also be written as
10038161003816100381610038161198861198991003816100381610038161003816 le
10038161003816100381610038161003816100381610038161003816
1198871199011199021
119887 (1 minus 119887) + 119887119901 minus 119901 + 119899
10038161003816100381610038161003816100381610038161003816
119899minus1
sum
119896=0
10038161003816100381610038161198861198961003816100381610038161003816 (48)
This concludes the proof
Competing Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work here is supported by AP-2013-009
References
[1] S Kanas and A Wisniowska ldquoConic domains and k-starlikefunctionsrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 45 no 4 pp 647ndash657 2000
[2] J-L Liu and H M Srivastava ldquoClasses of meromorphicallymultivalent functions associated with the generalized hyperge-ometric functionrdquo Mathematical and Computer Modelling vol39 no 1 pp 21ndash34 2004
[3] Z Shareef Some geometric properties of certain classes of analyticfunctions [PhD thesis] Universiti KebangsaanMalaysia BangiMalaysia 2015
[4] P Eenigenburg P TMocanu S SMiller andMO Reade ldquoOna briot-bouquet differential subordinationrdquo in General Inequal-ities vol 3 of International Series of Numerical Mathematics pp339ndash348 Birkhauser Basel Switzerland 1983
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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
The Scientific World Journal 7
[5] W Rogosinski ldquoOn the coefficients of subordinate functionsrdquoProceedings of the LondonMathematical Society Series 2 vol 48no 1 pp 48ndash82 1945
[6] F M Al-Oboudi and K A Al-Amoudi ldquoOn classes of analyticfunctions related to conic domainsrdquo Journal of MathematicalAnalysis and Applications vol 339 no 1 pp 655ndash667 2008
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