Coefficient estimates and subordination properties associated with certain classes...
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COEFFICIENT ESTIMATES AND SUBORDINATION PROPERTIES ASSOCIATED WITH CERTAIN CLASSES OF FUNCTIONS R. K. RAINA AND DEEPAK BANSAL Received 8 June 2005 and in revised form 13 September 2005 Dedicated to Professor H. M. Srivastava, University of Victoria, Canada, on his 65th Birth Anniversary This note investigates coefficient estimates and subordination properties for certain classes of normalized functions (which are essentially defined by means of a Hadamard product of two analytic functions). We exhibit several interesting consequences of our main results, and in the process, we are also led to the corrected forms of the results given by Owa and Nishiwaki (2002). 1. Introduction and preliminaries Let denote the class of functions f (z) normalized by f (0) = f (0) − 1 = 0, and analytic in the open unit disk ={z : z ∈ C, |z| < 1}; then f (z) can be expressed as f (z) = z + ∞ n=2 a n z n . (1.1) Let us denote by (α) and (α) two subclasses of the class , which are defined (for α> 1) as follows: (α) = f : f ∈ ; zf (z) f (z) <α (z ∈ ; α> 1) , (α) = f : f ∈ ; 1+ zf (z) f (z) <α (z ∈ ; α> 1) . (1.2) The classes (α) and (α) were studied recently by Owa and Nishiwaki [3] and also by Owa and Srivastava [4]. In fact, for 1 <α 4/ 3, these classes were investigated earlier by Uralegaddi et al. [7]. It follows from (1.2) that f (z) ∈ (α) ⇐⇒ zf (z) ∈ (α). (1.3) Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:22 (2005) 3685–3695 DOI: 10.1155/IJMMS.2005.3685
Transcript of Coefficient estimates and subordination properties associated with certain classes...
COEFFICIENT ESTIMATES AND SUBORDINATIONPROPERTIES ASSOCIATED WITH CERTAINCLASSES OF FUNCTIONS
R K RAINA AND DEEPAK BANSAL
Received 8 June 2005 and in revised form 13 September 2005
Dedicated to Professor H M Srivastava University of Victoria Canadaon his 65th Birth Anniversary
This note investigates coefficient estimates and subordination properties for certainclasses of normalized functions (which are essentially defined by means of a Hadamardproduct of two analytic functions) We exhibit several interesting consequences of ourmain results and in the process we are also led to the corrected forms of the results givenby Owa and Nishiwaki (2002)
1 Introduction and preliminaries
Let denote the class of functions f (z) normalized by f (0)= f prime(0)minus 1= 0 and analyticin the open unit disk = z z isin C |z| lt 1 then f (z) can be expressed as
f (z)= z+infinsumn=2
anzn (11)
Let us denote by (α) and (α) two subclasses of the class which are defined (forα gt 1) as follows
(α)=f f isin
(z f prime(z)f (z)
)lt α (z isin α gt 1)
(α)=f f isin
(1 +
z f primeprime(z)f prime(z)
)lt α (z isin α gt 1)
(12)
The classes (α) and (α) were studied recently by Owa and Nishiwaki [3] and also byOwa and Srivastava [4] In fact for 1 lt α 43 these classes were investigated earlier byUralegaddi et al [7]
It follows from (12) that
f (z)isin(α)lArrrArr z f prime(z)isin(α) (13)
Copyright copy 2005 Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical Sciences 200522 (2005) 3685ndash3695DOI 101155IJMMS20053685
3686 Coefficient and subordination properties
If f g isin where f (z) is given by (11) and g(z) is defined by
g(z)= z+infinsumn=2
bnzn (14)
then their Hadamard product (or convolution) f lowast g is defined (as usual) by
( f lowast g)(z)= z+infinsumn=2
anbnzn = (g lowast f )(z) (15)
For two functions f and g analytic in we say that the function f is subordinate to g in (denoted by f ≺ g) if there exists a Schwarz functionw(z) analytic in withw(0)= 0and |w(z)| lt 1 (z isin) such that f (z)= g(w(z))
We introduce here a class α(φψ) which is defined as follows suppose the functionsφ(z) and ψ(z) are given by
φ(z)= z+infinsumn=2
λnzn
ψ(z)= z+infinsumn=2
micronzn
(16)
where λn micron 0 (for all n isin N1) We say that f isin is in α(φψ) provided that( f lowastψ)(z) = 0 and
( f lowastφ)(z)( f lowastψ)(z)
lt α (α gt 1 z isin) (17)
Several new and known subclasses can be obtained from the class α(φψ) by suitablychoosing the functions φ(z) and ψ(z) We mention below some of these subclasses ofα(φψ) consisting of functions f (z)isin The Ruscheweyh derivative operator (see [5])micro rarr is defined by
microf (z)
= z
(1minus z)1+micro lowast f (z)(micro gtminus1 f (z)isin
) (18)
Indeed we have
α
(z
(1minus z)λ+2
z
(1minus z)λ+1
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)lt α (z isin α gt 1 λ gtminus1)
equivΛα(λ)
(19)
R K Raina and D Bansal 3687
and the relationships
α
(z
(1minus z)2
z
(1minus z)
)equiv(α) (α gt 1)
α
(z+ z2
(1minus z)3
z
(1minus z)2
)equiv(α) (α gt 1)
(110)
A similar type of a subclass Λα(λ) involving the Ruscheweyh operator Dλ (defined by(19) above) was also considered by Ali et al [1] whereas the subclasses (α) and (α)are the known classes (see [3]) defined by (12) respectively
In our present investigation we require the following definition and also a relatedresult due to Wilf [8]
Definition 11 A sequence bninfin1 of complex numbers is said to be a subordination factorsequence if whenever f (z) given by (11) is univalent and convex in then
infinsumn=1
bnanzn ≺ f (z) in (111)
Lemma 12 The sequence bninfin1 is a subordinating factor sequence if and only if
[
1 + 2infinsumn=1
bnzn
]gt 0 (z isin) (112)
The main purpose of this note is first to investigate the coefficient estimates and relatedproperties of certain classes of normalized functions defined as a Hadamard product (orconvolution) of two analytic functions in the open unit disk Additionally by defininganother analogous class of analytic functions we develop a subordination theorem forthis class and consider various interesting consequences (including the corrected formsof the results obtained in [3]) from our main results
2 Coefficient inequalities and related properties
We first derive a sufficient condition for the function f (z) to belong to the aforemen-tioned class α(φψ) The result is contained in the following
Theorem 21 If f (z)isin satisfies
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣∣∣an∣∣ 2(αminus 1) (21)
for some k (0 k 1) and some α (α gt 1) then f (z)isinα(φψ)
Proof Let condition (21) be satisfied for the function f (z)isin It is sufficient to showthat ∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k
( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣ lt 1 (z isin) (22)
3688 Coefficient and subordination properties
We note that
∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣=∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)anznminus1
(1minus 2α+ k) +suminfin
n=2
λnminus (2αminus k)micron
anznminus1
∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣|z|nminus1
(2αminus kminus 1)minussuminfinn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣|z|nminus1
lt(1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣(2αminus kminus 1)minussuminfin
n=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣
(23)
The extreme-right-side expression of the above inequality would remain bounded by 1 if
(1minus k) +infinsumn=2
(λnminus kmicron
)∣∣an∣∣ (2αminus kminus 1)minusinfinsumn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣ (24)
which leads to the desired inequality (21) This completes the proof
Taking k = 1 in (21) we observe that the sequence
ωn(α)= λn + (1minus 2α)micron(α gt 1 λn gt micron gt 0 nisinN1) (25)
remains nonnegative whenever the sequences 〈micron〉 and 〈λnmicron〉 are nondecreasing and αsatisfies the inequality 1 lt α (12)(1 + λ2micro2) Using this in Theorem 21 we obtain thefollowing
Corollary 22 If f (z)isin satisfies
infinsumn=2
(λnminusαmicron
)∣∣an∣∣ (αminus 1) (26)
provided that λn gt micron gt 0 〈micron〉 and 〈λnmicron〉 are nondecreasing sequences and α is such that1 lt α (12)(1 + λ2micro2) then f (z)isinα(φψ)
By appealing to (19) (110) when the functions φ(z) and ψ(z) in (17) are selectedappropriately then Theorem 21 yields the following results
Corollary 23 If f (z)isin satisfies
infinsumn=2
Γ(λ+n)Γ(λ+ 2)Γ(n)
[n+ (1minus k)λminus k+
∣∣n+ λ(1 + kminus 2α) + kminus 2α∣∣]∣∣an∣∣ 2(αminus 1)
(27)
for some k (0 k 1) and some α (α gt 1) then f (z)isinΛα(λ)
R K Raina and D Bansal 3689
Corollary 24 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|∣∣an∣∣ 2(αminus 1) (28)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Corollary 25 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|n∣∣an∣∣ 2(αminus 1) (29)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Remark 26 Corollary 23 is a new result whereas Corollaries 24 and 25 are the sameassertions as obtained earlier by Owa and Nishiwaki [3 Theorems 21 and 23 pages 2-3]
We next mention here another known class E(φψβ) (see [2]) consisting of the func-tion f (z)isin defined analogously to the class α(φψ) which satisfies the condition
( f lowastφ)(z)( f lowastψ)(z)
gt β (0 β lt 1 z isin) (210)
We prove the following result
Theorem 27 If f (z) isin satisfies (26) for some α (1 lt α 2λ2(λ2 + micro2)) then f (z) isinE(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Proof If α satisfies the inequality (1 lt α (12)(1 + λ2micro2)) then (26) of Corollary 22implies
infinsumn=2
(λnminusαmicron
)(αminus 1)
∣∣an∣∣ 1 (211)
Now using the following result [2 Theorem 1 page 72] which determines the sufficiencycondition for the function f (z)isin to be in the class E(φψβ)
infinsumn=2
(λnminusβmicron
)(1minusβ)
∣∣an∣∣ 1 (0 β lt 1) (212)
we need to find the smallest positive β such that
infinsumn=2
λnminusβmicron1minusβ
∣∣an∣∣infinsumn=2
λnminusαmicronαminus 1
∣∣an∣∣ 1 (213)
This gives
β 2λnminusαmicronminusαλnmicron + λnminus 2αmicron
(nisinN1) (214)
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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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
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Stochastic AnalysisInternational Journal of
3686 Coefficient and subordination properties
If f g isin where f (z) is given by (11) and g(z) is defined by
g(z)= z+infinsumn=2
bnzn (14)
then their Hadamard product (or convolution) f lowast g is defined (as usual) by
( f lowast g)(z)= z+infinsumn=2
anbnzn = (g lowast f )(z) (15)
For two functions f and g analytic in we say that the function f is subordinate to g in (denoted by f ≺ g) if there exists a Schwarz functionw(z) analytic in withw(0)= 0and |w(z)| lt 1 (z isin) such that f (z)= g(w(z))
We introduce here a class α(φψ) which is defined as follows suppose the functionsφ(z) and ψ(z) are given by
φ(z)= z+infinsumn=2
λnzn
ψ(z)= z+infinsumn=2
micronzn
(16)
where λn micron 0 (for all n isin N1) We say that f isin is in α(φψ) provided that( f lowastψ)(z) = 0 and
( f lowastφ)(z)( f lowastψ)(z)
lt α (α gt 1 z isin) (17)
Several new and known subclasses can be obtained from the class α(φψ) by suitablychoosing the functions φ(z) and ψ(z) We mention below some of these subclasses ofα(φψ) consisting of functions f (z)isin The Ruscheweyh derivative operator (see [5])micro rarr is defined by
microf (z)
= z
(1minus z)1+micro lowast f (z)(micro gtminus1 f (z)isin
) (18)
Indeed we have
α
(z
(1minus z)λ+2
z
(1minus z)λ+1
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)lt α (z isin α gt 1 λ gtminus1)
equivΛα(λ)
(19)
R K Raina and D Bansal 3687
and the relationships
α
(z
(1minus z)2
z
(1minus z)
)equiv(α) (α gt 1)
α
(z+ z2
(1minus z)3
z
(1minus z)2
)equiv(α) (α gt 1)
(110)
A similar type of a subclass Λα(λ) involving the Ruscheweyh operator Dλ (defined by(19) above) was also considered by Ali et al [1] whereas the subclasses (α) and (α)are the known classes (see [3]) defined by (12) respectively
In our present investigation we require the following definition and also a relatedresult due to Wilf [8]
Definition 11 A sequence bninfin1 of complex numbers is said to be a subordination factorsequence if whenever f (z) given by (11) is univalent and convex in then
infinsumn=1
bnanzn ≺ f (z) in (111)
Lemma 12 The sequence bninfin1 is a subordinating factor sequence if and only if
[
1 + 2infinsumn=1
bnzn
]gt 0 (z isin) (112)
The main purpose of this note is first to investigate the coefficient estimates and relatedproperties of certain classes of normalized functions defined as a Hadamard product (orconvolution) of two analytic functions in the open unit disk Additionally by defininganother analogous class of analytic functions we develop a subordination theorem forthis class and consider various interesting consequences (including the corrected formsof the results obtained in [3]) from our main results
2 Coefficient inequalities and related properties
We first derive a sufficient condition for the function f (z) to belong to the aforemen-tioned class α(φψ) The result is contained in the following
Theorem 21 If f (z)isin satisfies
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣∣∣an∣∣ 2(αminus 1) (21)
for some k (0 k 1) and some α (α gt 1) then f (z)isinα(φψ)
Proof Let condition (21) be satisfied for the function f (z)isin It is sufficient to showthat ∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k
( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣ lt 1 (z isin) (22)
3688 Coefficient and subordination properties
We note that
∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣=∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)anznminus1
(1minus 2α+ k) +suminfin
n=2
λnminus (2αminus k)micron
anznminus1
∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣|z|nminus1
(2αminus kminus 1)minussuminfinn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣|z|nminus1
lt(1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣(2αminus kminus 1)minussuminfin
n=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣
(23)
The extreme-right-side expression of the above inequality would remain bounded by 1 if
(1minus k) +infinsumn=2
(λnminus kmicron
)∣∣an∣∣ (2αminus kminus 1)minusinfinsumn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣ (24)
which leads to the desired inequality (21) This completes the proof
Taking k = 1 in (21) we observe that the sequence
ωn(α)= λn + (1minus 2α)micron(α gt 1 λn gt micron gt 0 nisinN1) (25)
remains nonnegative whenever the sequences 〈micron〉 and 〈λnmicron〉 are nondecreasing and αsatisfies the inequality 1 lt α (12)(1 + λ2micro2) Using this in Theorem 21 we obtain thefollowing
Corollary 22 If f (z)isin satisfies
infinsumn=2
(λnminusαmicron
)∣∣an∣∣ (αminus 1) (26)
provided that λn gt micron gt 0 〈micron〉 and 〈λnmicron〉 are nondecreasing sequences and α is such that1 lt α (12)(1 + λ2micro2) then f (z)isinα(φψ)
By appealing to (19) (110) when the functions φ(z) and ψ(z) in (17) are selectedappropriately then Theorem 21 yields the following results
Corollary 23 If f (z)isin satisfies
infinsumn=2
Γ(λ+n)Γ(λ+ 2)Γ(n)
[n+ (1minus k)λminus k+
∣∣n+ λ(1 + kminus 2α) + kminus 2α∣∣]∣∣an∣∣ 2(αminus 1)
(27)
for some k (0 k 1) and some α (α gt 1) then f (z)isinΛα(λ)
R K Raina and D Bansal 3689
Corollary 24 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|∣∣an∣∣ 2(αminus 1) (28)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Corollary 25 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|n∣∣an∣∣ 2(αminus 1) (29)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Remark 26 Corollary 23 is a new result whereas Corollaries 24 and 25 are the sameassertions as obtained earlier by Owa and Nishiwaki [3 Theorems 21 and 23 pages 2-3]
We next mention here another known class E(φψβ) (see [2]) consisting of the func-tion f (z)isin defined analogously to the class α(φψ) which satisfies the condition
( f lowastφ)(z)( f lowastψ)(z)
gt β (0 β lt 1 z isin) (210)
We prove the following result
Theorem 27 If f (z) isin satisfies (26) for some α (1 lt α 2λ2(λ2 + micro2)) then f (z) isinE(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Proof If α satisfies the inequality (1 lt α (12)(1 + λ2micro2)) then (26) of Corollary 22implies
infinsumn=2
(λnminusαmicron
)(αminus 1)
∣∣an∣∣ 1 (211)
Now using the following result [2 Theorem 1 page 72] which determines the sufficiencycondition for the function f (z)isin to be in the class E(φψβ)
infinsumn=2
(λnminusβmicron
)(1minusβ)
∣∣an∣∣ 1 (0 β lt 1) (212)
we need to find the smallest positive β such that
infinsumn=2
λnminusβmicron1minusβ
∣∣an∣∣infinsumn=2
λnminusαmicronαminus 1
∣∣an∣∣ 1 (213)
This gives
β 2λnminusαmicronminusαλnmicron + λnminus 2αmicron
(nisinN1) (214)
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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R K Raina and D Bansal 3687
and the relationships
α
(z
(1minus z)2
z
(1minus z)
)equiv(α) (α gt 1)
α
(z+ z2
(1minus z)3
z
(1minus z)2
)equiv(α) (α gt 1)
(110)
A similar type of a subclass Λα(λ) involving the Ruscheweyh operator Dλ (defined by(19) above) was also considered by Ali et al [1] whereas the subclasses (α) and (α)are the known classes (see [3]) defined by (12) respectively
In our present investigation we require the following definition and also a relatedresult due to Wilf [8]
Definition 11 A sequence bninfin1 of complex numbers is said to be a subordination factorsequence if whenever f (z) given by (11) is univalent and convex in then
infinsumn=1
bnanzn ≺ f (z) in (111)
Lemma 12 The sequence bninfin1 is a subordinating factor sequence if and only if
[
1 + 2infinsumn=1
bnzn
]gt 0 (z isin) (112)
The main purpose of this note is first to investigate the coefficient estimates and relatedproperties of certain classes of normalized functions defined as a Hadamard product (orconvolution) of two analytic functions in the open unit disk Additionally by defininganother analogous class of analytic functions we develop a subordination theorem forthis class and consider various interesting consequences (including the corrected formsof the results obtained in [3]) from our main results
2 Coefficient inequalities and related properties
We first derive a sufficient condition for the function f (z) to belong to the aforemen-tioned class α(φψ) The result is contained in the following
Theorem 21 If f (z)isin satisfies
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣∣∣an∣∣ 2(αminus 1) (21)
for some k (0 k 1) and some α (α gt 1) then f (z)isinα(φψ)
Proof Let condition (21) be satisfied for the function f (z)isin It is sufficient to showthat ∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k
( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣ lt 1 (z isin) (22)
3688 Coefficient and subordination properties
We note that
∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣=∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)anznminus1
(1minus 2α+ k) +suminfin
n=2
λnminus (2αminus k)micron
anznminus1
∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣|z|nminus1
(2αminus kminus 1)minussuminfinn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣|z|nminus1
lt(1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣(2αminus kminus 1)minussuminfin
n=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣
(23)
The extreme-right-side expression of the above inequality would remain bounded by 1 if
(1minus k) +infinsumn=2
(λnminus kmicron
)∣∣an∣∣ (2αminus kminus 1)minusinfinsumn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣ (24)
which leads to the desired inequality (21) This completes the proof
Taking k = 1 in (21) we observe that the sequence
ωn(α)= λn + (1minus 2α)micron(α gt 1 λn gt micron gt 0 nisinN1) (25)
remains nonnegative whenever the sequences 〈micron〉 and 〈λnmicron〉 are nondecreasing and αsatisfies the inequality 1 lt α (12)(1 + λ2micro2) Using this in Theorem 21 we obtain thefollowing
Corollary 22 If f (z)isin satisfies
infinsumn=2
(λnminusαmicron
)∣∣an∣∣ (αminus 1) (26)
provided that λn gt micron gt 0 〈micron〉 and 〈λnmicron〉 are nondecreasing sequences and α is such that1 lt α (12)(1 + λ2micro2) then f (z)isinα(φψ)
By appealing to (19) (110) when the functions φ(z) and ψ(z) in (17) are selectedappropriately then Theorem 21 yields the following results
Corollary 23 If f (z)isin satisfies
infinsumn=2
Γ(λ+n)Γ(λ+ 2)Γ(n)
[n+ (1minus k)λminus k+
∣∣n+ λ(1 + kminus 2α) + kminus 2α∣∣]∣∣an∣∣ 2(αminus 1)
(27)
for some k (0 k 1) and some α (α gt 1) then f (z)isinΛα(λ)
R K Raina and D Bansal 3689
Corollary 24 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|∣∣an∣∣ 2(αminus 1) (28)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Corollary 25 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|n∣∣an∣∣ 2(αminus 1) (29)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Remark 26 Corollary 23 is a new result whereas Corollaries 24 and 25 are the sameassertions as obtained earlier by Owa and Nishiwaki [3 Theorems 21 and 23 pages 2-3]
We next mention here another known class E(φψβ) (see [2]) consisting of the func-tion f (z)isin defined analogously to the class α(φψ) which satisfies the condition
( f lowastφ)(z)( f lowastψ)(z)
gt β (0 β lt 1 z isin) (210)
We prove the following result
Theorem 27 If f (z) isin satisfies (26) for some α (1 lt α 2λ2(λ2 + micro2)) then f (z) isinE(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Proof If α satisfies the inequality (1 lt α (12)(1 + λ2micro2)) then (26) of Corollary 22implies
infinsumn=2
(λnminusαmicron
)(αminus 1)
∣∣an∣∣ 1 (211)
Now using the following result [2 Theorem 1 page 72] which determines the sufficiencycondition for the function f (z)isin to be in the class E(φψβ)
infinsumn=2
(λnminusβmicron
)(1minusβ)
∣∣an∣∣ 1 (0 β lt 1) (212)
we need to find the smallest positive β such that
infinsumn=2
λnminusβmicron1minusβ
∣∣an∣∣infinsumn=2
λnminusαmicronαminus 1
∣∣an∣∣ 1 (213)
This gives
β 2λnminusαmicronminusαλnmicron + λnminus 2αmicron
(nisinN1) (214)
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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3688 Coefficient and subordination properties
We note that
∣∣∣∣∣ ( f lowastφ)(z)( f lowastψ)(z)minus k( f lowastφ)(z)( f lowastψ)(z)minus (2αminus k)
∣∣∣∣∣=∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)anznminus1
(1minus 2α+ k) +suminfin
n=2
λnminus (2αminus k)micron
anznminus1
∣∣∣∣∣ (1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣|z|nminus1
(2αminus kminus 1)minussuminfinn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣|z|nminus1
lt(1minus k) +
suminfinn=2
(λnminus kmicron
)∣∣an∣∣(2αminus kminus 1)minussuminfin
n=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣
(23)
The extreme-right-side expression of the above inequality would remain bounded by 1 if
(1minus k) +infinsumn=2
(λnminus kmicron
)∣∣an∣∣ (2αminus kminus 1)minusinfinsumn=2
∣∣λnminus (2αminus k)micron∣∣∣∣an∣∣ (24)
which leads to the desired inequality (21) This completes the proof
Taking k = 1 in (21) we observe that the sequence
ωn(α)= λn + (1minus 2α)micron(α gt 1 λn gt micron gt 0 nisinN1) (25)
remains nonnegative whenever the sequences 〈micron〉 and 〈λnmicron〉 are nondecreasing and αsatisfies the inequality 1 lt α (12)(1 + λ2micro2) Using this in Theorem 21 we obtain thefollowing
Corollary 22 If f (z)isin satisfies
infinsumn=2
(λnminusαmicron
)∣∣an∣∣ (αminus 1) (26)
provided that λn gt micron gt 0 〈micron〉 and 〈λnmicron〉 are nondecreasing sequences and α is such that1 lt α (12)(1 + λ2micro2) then f (z)isinα(φψ)
By appealing to (19) (110) when the functions φ(z) and ψ(z) in (17) are selectedappropriately then Theorem 21 yields the following results
Corollary 23 If f (z)isin satisfies
infinsumn=2
Γ(λ+n)Γ(λ+ 2)Γ(n)
[n+ (1minus k)λminus k+
∣∣n+ λ(1 + kminus 2α) + kminus 2α∣∣]∣∣an∣∣ 2(αminus 1)
(27)
for some k (0 k 1) and some α (α gt 1) then f (z)isinΛα(λ)
R K Raina and D Bansal 3689
Corollary 24 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|∣∣an∣∣ 2(αminus 1) (28)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Corollary 25 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|n∣∣an∣∣ 2(αminus 1) (29)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Remark 26 Corollary 23 is a new result whereas Corollaries 24 and 25 are the sameassertions as obtained earlier by Owa and Nishiwaki [3 Theorems 21 and 23 pages 2-3]
We next mention here another known class E(φψβ) (see [2]) consisting of the func-tion f (z)isin defined analogously to the class α(φψ) which satisfies the condition
( f lowastφ)(z)( f lowastψ)(z)
gt β (0 β lt 1 z isin) (210)
We prove the following result
Theorem 27 If f (z) isin satisfies (26) for some α (1 lt α 2λ2(λ2 + micro2)) then f (z) isinE(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Proof If α satisfies the inequality (1 lt α (12)(1 + λ2micro2)) then (26) of Corollary 22implies
infinsumn=2
(λnminusαmicron
)(αminus 1)
∣∣an∣∣ 1 (211)
Now using the following result [2 Theorem 1 page 72] which determines the sufficiencycondition for the function f (z)isin to be in the class E(φψβ)
infinsumn=2
(λnminusβmicron
)(1minusβ)
∣∣an∣∣ 1 (0 β lt 1) (212)
we need to find the smallest positive β such that
infinsumn=2
λnminusβmicron1minusβ
∣∣an∣∣infinsumn=2
λnminusαmicronαminus 1
∣∣an∣∣ 1 (213)
This gives
β 2λnminusαmicronminusαλnmicron + λnminus 2αmicron
(nisinN1) (214)
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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R K Raina and D Bansal 3689
Corollary 24 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|∣∣an∣∣ 2(αminus 1) (28)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Corollary 25 If f (z)isin satisfies
infinsumn=2
(nminus k) + |n+ kminus 2α|n∣∣an∣∣ 2(αminus 1) (29)
for some k (0 k 1) and some α (α gt 1) then f (z)isin(α)
Remark 26 Corollary 23 is a new result whereas Corollaries 24 and 25 are the sameassertions as obtained earlier by Owa and Nishiwaki [3 Theorems 21 and 23 pages 2-3]
We next mention here another known class E(φψβ) (see [2]) consisting of the func-tion f (z)isin defined analogously to the class α(φψ) which satisfies the condition
( f lowastφ)(z)( f lowastψ)(z)
gt β (0 β lt 1 z isin) (210)
We prove the following result
Theorem 27 If f (z) isin satisfies (26) for some α (1 lt α 2λ2(λ2 + micro2)) then f (z) isinE(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Proof If α satisfies the inequality (1 lt α (12)(1 + λ2micro2)) then (26) of Corollary 22implies
infinsumn=2
(λnminusαmicron
)(αminus 1)
∣∣an∣∣ 1 (211)
Now using the following result [2 Theorem 1 page 72] which determines the sufficiencycondition for the function f (z)isin to be in the class E(φψβ)
infinsumn=2
(λnminusβmicron
)(1minusβ)
∣∣an∣∣ 1 (0 β lt 1) (212)
we need to find the smallest positive β such that
infinsumn=2
λnminusβmicron1minusβ
∣∣an∣∣infinsumn=2
λnminusαmicronαminus 1
∣∣an∣∣ 1 (213)
This gives
β 2λnminusαmicronminusαλnmicron + λnminus 2αmicron
(nisinN1) (214)
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
3690 Coefficient and subordination properties
Let us put
F(n)= 2λnminusαmicronminusαλnmicron + λnminus 2αmicron(
λn gt micron gt 0 nisinN1 1 lt α lt12
(1 +
λ2
micro2
)langλnmicron
is nondecreasing
)
(215)
We will show that F(n) is a nondecreasing function of n Elementary calculations give
F(n+ 1)minusF(n)= 2(αminus 1)2(λn+1micronminusmicron+1λn
)ωn+1(α)ωn(α)
(216)
which is observed to be positive under the constraints stated with (215) where ωn(α)given by (25) is a nonvanishing sequence Hence we conclude from (214) that
β 2λ2minusαmicro2minusαλ2
micro2 + λ2minus 2αmicro2 (217)
which proves that f (z)isin E(φψβ) (β = (2λ2minusαmicro2minusαλ2)(micro2 + λ2minus 2αmicro2))
Setting β = 12 in (210) and choosing φ(z) and ψ(z) appropriately (as mentioned in(19)) we obtain the class Kλ involving the Ruschweyh derivative Dλ (see [5]) which isdefined by
E
(z
(1minus z)λ+2
z
(1minus z)λ+112
)
=f f isin
(Dλ+1 f (z)Dλ f (z)
)gt
12
(z isin λ gtminus1)
equivKλ
(218)
Also the class E(φψβ) includes among others the familiar subclasses of which con-sist of starlike functions lowast(β) of order β (0 β lt 1) and convex functions (β) oforder β (0 β lt 1) If we select the functions φ(z) and ψ(z) according to (19) (110) inTheorem 27 then we obtain the following results
Corollary 28 If f (z) isin satisfies the coefficient inequality (27) for k = 1 and some α(1 lt α (2λ+ 5)2(λ+ 2)) then f (z)isinKλ
Corollary 29 If f (z) isin satisfies the coefficient inequality (28) for k = 1 and some α(1 lt α 43) then f (z)isinlowast(β) (β = (4minus 3α)(3minus 2α))
Corollary 210 If f (z)isin satisfies the coefficient inequality (29) for k = 1 and some α(1 lt α 43) then f (z)isin(β) (β = (4minus 3α)(3minus 2α))
Remark 211 Corollary 28 is a new result and Corollaries 29 and 210 are the correctedforms of the known assertions stated in [3 Theorem 25 page 3]
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
R K Raina and D Bansal 3691
3 Subordination theorem
Before stating and proving our subordination theorem we define a class Gα(φψ) as fol-lows
We denote Gα(φψ) to be a class of functions f isin α(φψ) whose coefficients satisfythe condition (21) Evidently we have
Gα
(z
(1minus z)λ+2
z
(1minus z)λ+1
)equivΛlowastα (λ)
Gα
(z
(1minus z)2
z
1minus z
)equivlowast(α)
Gα
(z+ z2
(1minus z)3
z
(1minus z)2
)equivlowast(α)
(31)
where Λlowastα (λ) lowast(α) and lowast(α) are respectively the subclasses of Λα(λ) (α) and(α) consisting of functions f (z) isin satisfying inequalities (27) (28) and (29) re-spectively Again it is obvious that
Λlowastα (λ)subΛα(λ) lowast(α)sub(α) lowast(α)sub(α) (32)
Theorem 31 Let f isin Gα(φψ) and let the sequences 〈λn〉 〈micron〉 and 〈λnmicron〉 be nonde-creasing then
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)≺ g(z)
(λn gt micron gt 0 α gt 1 0 k 1 z isin
) (33)
for every function g isin In particular
f (z)gtminus
(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ (z isin) (34)
The constant factor
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ (35)
in the subordination result (33) cannot be replaced by any larger one
Proof Let f (z) defined by (11) belong to the class Gα(φψ) and let g(z) defined by (14)be any function in the class It follows then that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ ( f lowast g)(z)
=(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣(z+
infinsumn=2
anbnzn
)
(36)
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
3692 Coefficient and subordination properties
By invoking Definition 11 the subordination (33) of our theorem will hold true if thesequence
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2
(2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣aninfinn=1
(37)
is a subordinating factor sequence By virtue of Lemma 12 this is equivalent to the in-equality
(
1 + 2infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)gt 0 (z isin) (38)
Let us put
σ(n)= (λnminus kmicron)+∣∣λn + (kminus 2α)micron
∣∣ (39)
and we write
σ(n)= micron(
λnmicronminus k
)+
∣∣∣∣∣λnmicron + (kminus 2α)
∣∣∣∣∣ (310)
It is observed that the sequence σ(n) is a nondecreasing function of n under the con-straints
(λn micron gt 0 α gt 1 0 k 1 nisinN1) (311)
In particular (under the same condition)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣ (312)
Therefore for |z| = r (r lt 1) we obtain
(
1 +infinsumn=1
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
=(
1 +
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z+
infinsumn=2
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣anzn)
1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus
infinsumn=2
(λnminus kmicron
)+∣∣λn + (kminus 2α)micron
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣∣∣an∣∣rn
gt 1minus(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣(2αminus 2 + λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣ rminus 2(αminus 1)(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣ r gt 0
(313)
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
R K Raina and D Bansal 3693
This evidently establishes the inequality (38) and consequently the subordination rela-tion (33) of our Theorem 31 is proved The assertion (34) follows readily from (33)when the function g(z) is selected as
g(z)= z
1minus z = z+infinsumn=2
zn isin (314)
The sharpness of the multiplying factor in (33) can be established by considering a func-tion h(z) defined by
h(z)= zminus 2(αminus 1)(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣z2 (315)
which belongs to the class Gα(φψ) Using (33) we infer that
(λ2minus kmicro2
)+∣∣λ2 + (kminus 2α)micro2
∣∣2(
2αminus 2 + λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣h(z)≺ z
1minus z (316)
It can easily be verified that
min|z|1
( (
λ2minus kmicro2)
+∣∣λ2 + (kminus 2α)micro2
∣∣2
2αminus 2 + λ2minus kmicro2 +∣∣λ2 + (kminus 2α)micro2
∣∣h(z)
)=minus1
2 (317)
which completes the proof of Theorem 31
On choosing the arbitrary functions φ(z) and ψ(z) in accordance with the subclassesdefined by (31) we arrive at the following results
Corollary 32 Let f (z)isinΛlowastα (λ) then for every function g in
(λ(1minus k) + 2minus k)+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣
2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ ( f lowast g)(z)≺ g(z)
(α gt 1 0 k 1 λ gtminus1 z isin)
f (z)gtminus λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣(
λ(1minus k) + 2minus k)+∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣ (z isin)
(318)
The following constant factor
λ(1minus k) +∣∣λ(1 + kminus 2α) + 2 + kminus 2α
∣∣2λ(1minus k) + 2αminus k+
∣∣λ(1 + kminus 2α) + 2 + kminus 2α∣∣ (319)
cannot be replaced by a larger one
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
3694 Coefficient and subordination properties
Corollary 33 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (2αminus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(320)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(2αminus k) + |2 + kminus 2α| (321)
cannot be replaced by a larger one
Corollary 34 Let f (z)isinlowast(α) then for every function g in
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| ( f lowast g)(z)≺ g(z)
f (z)gtminus (α+ 1minus k) + |2 + kminus 2α|
(2minus k) + |2 + kminus 2α| (z isin)(322)
The following constant factor
(2minus k) + |2 + kminus 2α|2
(α+ 1minus k) + |2 + kminus 2α| (323)
cannot be replaced by a larger one
Remark 35 Corollary 32 is a new result and Corollaries 33 and 34 correspond to theknown results due to Srivastava and Attiya [6]
Acknowledgment
The authors would like to express their sincerest thanks to the referees for useful com-ments The present investigation was supported by AICTE (Government of India) NewDelhi
References
[1] R M Ali M Hussain Khan V Ravichandran and K G Subramanian A class of multivalentfunctions with positive coefficients defined by convolution JIPAM J Inequal Pure Appl Math6 (2005) no 1 art 22
[2] O P Juneja T R Reddy and M L Mogra A convolution approach for analytic functions withnegative coefficients Soochow J Math 11 (1985) 69ndash81
[3] S Owa and J Nishiwaki Coefficient estimates for certain classes of analytic functions JIPAM JInequal Pure Appl Math 3 (2002) no 5 art 72
[4] S Owa and H M Srivastava Some generalized convolution properties associated with certainsubclasses of analytic functions JIPAM J Inequal Pure Appl Math 3 (2002) no 3 art 42
[5] S Ruscheweyh New criteria for univalent functions Proc Amer Math Soc 49 (1975) no 1109ndash115
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
R K Raina and D Bansal 3695
[6] H M Srivastava and A A Attiya Some subordination results associated with certain subclassesof analytic functions JIPAM J Inequal Pure Appl Math 5 (2004) no 4 art 82
[7] B A Uralegaddi M D Ganigi and S M Sarangi Univalent functions with positive coefficientsTamkang J Math 25 (1994) no 3 225ndash230
[8] H S Wilf Subordinating factor sequences for convex maps of the unit circle Proc Amer MathSoc 12 (1961) no 5 689ndash693
R K Raina Department of Mathematics College of Technology and Engineering M P Universityof Agriculture and Technology Udaipur-313001 Rajasthan India
E-mail address rainark 7hotmailcom
Deepak Bansal Department of Mathematics College of Science M L Sukhadia UniversityUdaipur-313001 Rajasthan India
E-mail address deepakbansal 79yahoocom
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MathematicsJournal of
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
