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Abstract and Applied Analysis
Volume 2011, Article ID 726518, 9 pages
doi:10.1155/2011/726518
Research Article
Differential Subordinations for Certain
Meromorphically Multivalent Functions Defined
by Dziok-Srivastava Operator
Ying Yang,1 Yu-Qin Tao,1 and Jin-Lin Liu2
1
2
Department of Mathematics, Maanshan Teacher’s College, Maanshan 243000, China
Department of Mathematics, Yangzhou University, Yangzhou 225002, China
Correspondence should be addressed to Jin-Lin Liu, jlliu@yzu.edu.cn
Received 21 January 2011; Revised 14 April 2011; Accepted 20 April 2011
Academic Editor: Malisa R. Zizovic
Copyright q 2011 Ying Yang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
By making use of the Dziok-Srivastava operator, we introduce a new class of meromorphically
multivalent functions. Some inclusion properties of functions belonging to this class are derived.
1. Introduction
Let Σp denote the class of functions of the form
fz z−p ∞
an zn−p
p ∈ N : {1, 2, 3, . . .} ,
1.1
n1
which are analytic in the punctured open unit disk U0 {z : 0 < |z| < 1} with a pole at z 0.
Also let the Hadamard product or convolution of the following functions:
fj z z−p ∞
an,j zn−p
j 1, 2
1.2
n1
be given by
∞
f1 ∗ f2 z : z−p an,1 an,2 zn−p f2 ∗ f1 z.
n1
1.3
2
Abstract and Applied Analysis
Given two functions fz and gz, which are analytic in U U0 ∪ {0}, we say that the
function gz is subordinate to fz and write g ≺ f or more precisely gz ≺ fz z ∈ U,
if there exists a Schwarz function wz, analytic in U with w0 0 and |wz| < 1 z ∈
U such that gz fwz z ∈ U. In particular, if fz is univalent in U, we have the
following equivalence:
gz ≺ fz
z ∈ U ⇐⇒ g0 f0,
gU ⊂ fU.
1.4
Let A be the class of functions of the form
fz z ∞
bn zn ,
1.5
n2
which are analytic in U. A function fz ∈ A is said to be in the class S∗ α if
Re
zf z
fz
> α z ∈ U
1.6
for some α α < 1. When 0 ≤ α < 1, S∗ α is the class of starlike functions of order α in U. A
function fz ∈ A is said to be prestarlike of order α in U if
z
1 − z
21−α
∗ fz ∈ S∗ α
α < 1,
1.7
where the symbol ∗ means the familiar Hadamard product or convolution of two analytic
functions in U. We denote this class by Rα see 1. Clearly a function fz ∈ A is in the
class R0 if and only if fz is convex univalent in U and R1/2 S∗ 1/2.
For complex parameters
α1 , . . . , αq and β1 , . . . , βs
/ Z−0 : {0, −1, −2, . . .}; j 1, 2, . . . , s ,
βj ∈
1.8
we define the generalized hypergeometric function q Fs α1 , . . . , αq ; β1 , . . . , βs ; z by
∞ α · · · α
1 n
q
zn
n ·
q Fs α1 , . . . , αq ; β1 , . . . , βs ; z :
n!
n0 β1 n · · · βs n
q ≤ s 1; q, s ∈ N0 : N ∪ {0}; z ∈ U ,
1.9
where xn is the Pochhammer symbol defined, in terms of the Gamma function Γz,by
⎧
Γx n ⎨1
xn :
⎩
Γx
xx 1 · · · x n − 1
n 0; x ∈ C \ {0}
n ∈ N; x ∈ C.
1.10
Abstract and Applied Analysis
3
Corresponding to a function hp α1 , . . . , αq ; β1 , . . . , βs ; z defined by
hp α1 , . . . , αq ; β1 , . . . , βs ; z z−p q Fs α1 , . . . , αq ; β1 , . . . , βs ; z ,
1.11
we now consider a linear operator
Hp α1 , . . . , αq ; β1 , . . . , βs : Σ p −→ Σ p ,
1.12
defined by means of the Hadamard product or convolution as follows:
Hp α1 , . . . , αq ; β1 , . . . , βs fz : hp α1 , . . . , αq ; β1 , . . . , βs ; z ∗ fz.
1.13
For convenience, we write
Hp,q,s α1 : Hp α1 , . . . , αq ; β1 , . . . , βs .
1.14
Thus, after some calculations, we have
z Hp,q,s α1 fz α1 Hp,q,s α1 1fz − α1 p Hp,q,s α1 fz.
1.15
The operator Hp,q,s α1 is popularly known as the generalized Dziok-Srivastava
operator. Many interesting subclasses of multivalent functions, associated with the operator
Hp,q,s α1 and its various special cases, were investigated recently by e.g. Dziok and
Srivastava 2–4, Liu 5, Liu and Srivastava 6, 7, Patel et al. 8, Wang et al. 9, and others.
Let P be the class of functions hz with h0 1, which are analytic and convex
univalent in U.
Definition 1.1. A function fz ∈ Σp is said to be in the class Tp,q,s α1 , λ; h if it satisfies the
subordination condition
λ
λ − 1 p1 z
Hp,q,s α1 fz zp2 Hp,q,s α1 fz ≺ hz,
p
p p1
1.16
where λ is a complex number and hz ∈ P .
The main object of this paper is to present a systematic investigation of the
class Tp,q,s α1 , λ; h defined above by means of the generalized Dziok-Srivastava operator
Hp,q,s α1 .
For our purpose, we shall need the following lemmas to derive our main results for
the class Tp,q,s α1 , λ; h.
Lemma 1.2 see 10. Let gz be analytic in U and hz be analytic and convex univalent in U
with h0 g0. If
gz 1 zg z ≺ hz,
μ
1.17
4
Abstract and Applied Analysis
where Re μ > 0, then
gz ≺ hz
μz−μ
z
tμ−1 htdt ≺ hz
1.18
v
and hz
is the best dominant of 1.17.
Lemma 1.3 see 1. Let α < 1, fz ∈ S∗ α and gz ∈ Rα. Then, for any analytic function
Fz in U,
g ∗ fF
U ⊂ coFU,
g∗f
1.19
where coFU denotes the closed convex hull of FU.
2. Properties of the Class Tp,q,s α1 , λ; h
Theorem 2.1. Let λ1 < λ2 ≤ 0. Then Tp,q,s α1 , λ1 ; h ⊂ Tp,q,s α1 , λ2 ; h.
Proof. Let λ1 < λ2 ≤ 0 and suppose that
zp1 Hp,q,s α1 fz gz −
p
2.1
for fz ∈ Tp,q,s α1 , λ1 ; h. Then the function gz is analytic in U with g0 1. Differentiating
both sides of 2.1 with respect to z and using 1.16, we have
λ1
λ1
λ1 − 1 p1 z
zg z ≺ hz.
Hp,q,s α1 fz zp2 Hp,q,s α1 fz gz −
p
p1
p p1
2.2
Hence an application of Lemma 1.2 yields
gz ≺ hz.
2.3
Noting that 0 < λ2 /λ1 < 1 and that hz is convex univalent in U, it follows from 2.1 to 2.3
that
λ2
λ2 − 1 p1 z
Hp,q,s α1 fz zp2 Hp,q,s α1 fz
p
p p1
λ2 λ1 − 1 p1 λ1
p2
z
Hp,q,s α1 fz Hp,q,s α1 fz
z
λ1
p
p p1
λ2
1−
gz ≺ hz.
λ1
Thus fz ∈ T p, q, sα1 , λ2 ; h and the proof of Theorem 2.1 is completed.
2.4
Abstract and Applied Analysis
5
Theorem 2.2. Let 0 < b1 < b2 . Then Tp,q,s b2 , λ; h ⊂ Tp,q,s b1 , λ; h.
Proof. Define a function gz by
gz z ∞
b1 n
n1
b2 n
zn1
z ∈ U; 0 < b1 < b2 .
2.5
Then
zp1 hp b1 , α2 , . . . , αs , 1; b2 , α2 , . . . , αs ; z gz ∈ A,
2.6
hp b1 , α2 , . . . , αs , 1; b2 , α2 , . . . , αs ; z
2.7
where
is defined as in 1.11, and
z
1 − z
b2
∗ gz z
1 − zb1
2.8
.
By 2.8, we see that
z
1 − zb2
b1
b2
∗ gz ∈ S∗ 1 −
⊂ S∗ 1 −
2
2
0 < b1 < b2 ,
2.9
which implies that
b2
.
gz ∈ R 1 −
2
2.10
Let fz ∈ Tp,q,s b2 , λ; h. It is easy to verify that
zp1 Hp,q,s b1 fz zp hp b1 , α2 , . . . , αs , 1; b2 , α2 , . . . , αs ; z ∗ zp1 Hp,q,s b2 fz
2.11
zp2 Hp,q,s b1 fz zp hp b1 , α2 , . . . , αs , 1; b2 , α2 , . . . , αs ; z ∗ zp2 Hp,q,s b2 fz .
2.12
From 2.11, 2.12, and 2.6, we deduce that
λ
λ − 1 p1 z
Hp,q,s b1 fz zp2 Hp,q,s b1 fz
p
p p1
gz ∗ zwz
gz
∗ wz ,
z
gz ∗ z
2.13
6
Abstract and Applied Analysis
where
wz :
λ
λ − 1 p1 z
Hp,q,s b2 fz zp2 Hp,q,s b2 fz ≺ hz.
p
p p1
2.14
Since the function z belongs to the function class S∗ 1 − b2 /2 and hz is convex univalent in
U, it follows from 2.12, 2.13, 2.14, and Lemma 1.3 that
λ
λ − 1 p1 z
Hp,q,s b1 fz zp2 Hp,q,s b1 fz ≺ hz.
p
p p1
2.15
Thus fz ∈ Tp,q,s b1 , λ; h and the proof of Theorem 2.2 is completed.
Theorem 2.3. Let fz ∈ Tp,q,s α1 , λ; h, gz ∈ Σp and
1
Re zp gz >
2
z ∈ U.
2.16
f ∗ g z ∈ Tp,q,s α1 , λ; h.
2.17
Then
Proof. For fz ∈ Tp,q,s α1 , λ; h and gz ∈ Σp, we have
λ
λ − 1 p1 z
Hp,q,s α1 f ∗ g z zp2 Hp,q,s α1 f ∗ g z
p
p p1
λ
λ − 1 p
z gz ∗ zp1 Hp,q,s α1 fz zp gz ∗ zp2 Hp,q,s α1 fz
p
p p1
p
z gz ∗ ψz,
2.18
where
ψz λ
λ − 1 p1 z
Hp,q,s α1 fz zp2 Hp,q,s α1 fz .
p
p p1
2.19
In view of 2.16, the function zp gz has the Herglotz representation
z gz p
|x|1
dμx
1 − xz
z ∈ U,
2.20
where μx is a probability measure defined on the unit circle |x| 1 and
|x|1
dμx 1.
2.21
Abstract and Applied Analysis
7
Since hz is convex univalent in U, it follows from 2.18 to 2.20 that
λ
λ − 1 p1 z
Hp,q,s α1 f ∗ g z zp2 Hp,q,s α1 f ∗ g z
p
p p1
|x|1
2.22
ψxzdμx ≺ hz.
This shows that f ∗ gz ∈ Tp,q,s α1 , λ; h and the theorem is proved.
Theorem 2.4. Let fz ∈ Tp,q,s α1 , λ; h, gz ∈ Σp and
zp1 gz ∈ Rα
α < 1.
2.23
f ∗ g z ∈ Tp,q,s α1 , λ; h.
2.24
Then
Proof. For fz ∈ Tp,q,s α1 , λ; h and gz ∈ Σp, from 2.18 we have
λ
λ − 1 p1 z
Hp,q,s f ∗ g z zp2 Hp,q,s f ∗ g z
p
p p1
zp1 gz ∗ zψz
p1
z gz ∗ z
2.25
z ∈ U,
where ψz is defined as in 2.19.
Since hz is convex univalent in U,
zp1 gz ∈ Rα,
ψz ≺ hz,
z ∈ S∗ α α < 1,
2.26
it follows from 2.25 and Lemma 1.3 the desired result.
Theorem 2.5. Let λ < 0, β > 0 and fz ∈ Tp,q,s λ; βh 1 − β. If β ≤ β0 , where
1
β0 2
p1
1
λ
1
0
−1
u−p1/λ−1
du
1u
,
2.27
then fz ∈ Tp,q,s 0; h. The bound β0 is sharp when hz 1/1 − z.
Proof. Let us define
zp1 Hp,q,s α1 fz gz −
p
2.28
8
Abstract and Applied Analysis
for fz ∈ Tp,q,s λ; βh 1 − β with λ < 0 and β > 0. Then we have
gz −
λ
λ
λ − 1 p1 zg z z
Hp,q,s α1 fz zp2 Hp,q,s α1 fz
p1
p
p p1
2.29
≺ βhz 1 − β.
Hence an application of Lemma 1.2 yields
β p 1 p1/λ z −p1/λ−1
z
t
htdt 1 − β
λ
0
h ∗ ψ z,
gz ≺ −
2.30
where
β p 1 p1/λ z t−p1/λ−1
ψz −
z
dt 1 − β.
λ
1−t
0
2.31
If 0 < β ≤ β0 , where β0 > 1 is given by 2.27, then it follows from 2.31 that
β p 1 1 −p1/λ−1
1
Re ψz −
u
Re
du 1 − β
λ
1 − uz
0
β p 1 1 u−p1/λ−1
du 1 − β
>−
λ
1u
0
≥
1
2
2.32
z ∈ U; λ < 0.
Now, by using the Herglotz representation for ψz, from 2.28 and 2.30, we arrive at
zp1 Hp,q,s α1 fz ≺ h ∗ ψ z ≺ hz
−
p
2.33
because hz is convex univalent in U. This shows that fz ∈ Tp,q,s 0; h.
For hz 1/1 − z and fz ∈ Σp defined by
zp1 Hp,q,s α1 fz β p 1 p1/λ z t−p1/λ−1
−
z
dt 1 − β,
−
p
λ
1−t
0
2.34
it is easy to verify that
λ
λ − 1 p1 z
Hp,q,s α1 fz zp2 Hp,q,s α1 fz βhz 1 − β.
p
p p1
2.35
Abstract and Applied Analysis
9
Thus fz ∈ Tp,q,s λ; βh 1 − β. Also, for β > β0 , we have
zp1 Hp,q,s α1 fz β p 1 1 u−p1/λ−1
1
Re −
du 1 − β <
−→ −
p
λ
1u
2
0
z −→ −1,
2.36
which implies that fz ∈
/ Tp,q,s 0; h. Hence the bound β0 cannot be increased when hz 1/1 − z.
Acknowledgment
The authers would like to express sincere thanks to the referees for careful reading and
suggestions which helped us to improve the paper.
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