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Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator
Journal of Inequalities and Applications volume 2011, Article number: 486595 (2011)
Abstract
Subordination and superordination preserving properties for multivalent functions in the open unit disk associated with the Dziok-Srivastava operator are derived. Sandwich-type theorems for these multivalent functions are also obtained.
1. Introduction
Let be the open unit disk in the complex plane
, and let
denote the class of analytic functions defined in
For
and
, let
consist of functions
of the form
. Let
and
be members of
. The function
is said to be subordinate to
, or
is said to be superordinate to
, if there exists a function
analytic in
, with
and such that
. In such a case, we write
or
. If the function
is univalent in
, then
if and only if
and
(cf. [1, 2]). Let
, and let
be univalent in
. The subordination
is called a first-order differential subordination. It is of interest to determine conditions under which
arises for a prescribed univalent function
. The theory of differential subordination in
is a generalization of a differential inequality in
, and this theory of differential subordination was initiated by the works of Miller, Mocanu, and Reade in 1981. Recently, Miller and Mocanu [3] investigated the dual problem of differential superordination. The monograph by Miller and Mocanu [1] gives a good introduction to the theory of differential subordination, while the book by Bulboacă [4] investigates both subordination and superordination. Related results on superordination can be found in [5–23].
By using the theory of differential subordination, various subordination preserving properties for certain integral operators were obtained by Bulboacă [24], Miller et al. [25], and Owa and Srivastava [26]. The corresponding superordination properties and sandwich-type results were also investigated, for example, in [4]. In the present paper, we investigate subordination and superordination preserving properties of functions defined through the use of the Dziok-Srivastava linear operator (see (1.9) and (1.10)), and also obtain corresponding sandwich-type theorems.
The Dziok-Srivastava linear operator is a particular instance of a linear operator defined by convolution. For , let
denote the class of functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ1_HTML.gif)
that are analytic and -valent in the open unit disk
with
. The Hadamard product (or convolution)
of two analytic functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ2_HTML.gif)
is defined by the series
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ3_HTML.gif)
For complex parameters and
, the generalized hypergeometric function
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ4_HTML.gif)
where is the Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma function) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ5_HTML.gif)
To define the Dziok-Srivastava operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ6_HTML.gif)
via the Hadamard product given by (1.3), we consider a corresponding function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ7_HTML.gif)
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ8_HTML.gif)
The Dziok-Srivastava linear operator is now defined by the Hadamard product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ9_HTML.gif)
This operator was introduced and studied in a series of recent papers by Dziok and Srivastava ([27–29]; see also [30, 31]). For convenience, we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ10_HTML.gif)
The importance of the Dziok-Srivastava operator from the general convolution operator rests on the relation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ11_HTML.gif)
that can be verified by direct calculations (see, e.g., [27]). The linear operator includes various other linear operators as special cases. These include the operators introduced and studied by Carlson and Shaffer [32], Hohlov ([33], also see [34, 35]), and Ruscheweyh [36], as well as works in [27, 37].
2. Definitions and Lemmas
Recall that a domain is convex if the line segment joining any two points in
lies entirely in
, while the domain is starlike with respect to a point
if the line segment joining any point in
to
lies inside
. An analytic function
is convex or starlike if
is, respectively, convex or starlike with respect to 0. For
, analytically, these functions are described by the conditions
or
, respectively. More generally, for
, the classes of convex functions of order
and starlike functions of order
are, respectively, defined by
or
. A function
is close-to-convex if there is a convex function
(not necessarily normalized) such that
. Close-to-convex functions are known to be univalent.
The following definitions and lemmas will also be required in our present investigation.
Definition 2.1 (see [1, page 16]).
Let , and let
be univalent in
. If
is analytic in
and satisfies the differential subordination
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ12_HTML.gif)
then is called a solution of differential subordination (2.1). A univalent function
is called a dominant of the solutions of differential subordination (2.1), or more simply a dominant, if
for all
satisfying (2.1). A dominant
that satisfies
for all dominants
of (2.1) is said to be the best dominant of (2.1).
Definition 2.2 (see [3, Definition 1, pages 816-817]).
Let , and let
be analytic in
. If
and
are univalent in
and satisfy the differential superordination
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ13_HTML.gif)
then is called a solution of differential superordination (2.2). An analytic function
is called a subordinant of the solutions of differential superordination (2.2), or more simply a subordinant, if
for all
satisfying (2.2). A univalent subordinant
that satisfies
for all subordinants
of (2.2) is said to be the best subordinant of (2.2).
Definition 2.3 (see [1, Definition 2.2b, page 21]).
Denote by the class of functions
that are analytic and injective on
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ14_HTML.gif)
and are such that for
.
Lemma 2.4 (cf. [1, Theorem 2.3i, page 35]).
Suppose that the function satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ15_HTML.gif)
for all real and
, where
is a positive integer. If the function
is analytic in
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ16_HTML.gif)
then in
.
One of the points of importance of Lemma 2.4 was its use in showing that every convex function is starlike of order 1/2 (see e.g., [38, Theorem 2.6a, page 57]). In this paper, we take an opportunity to use the technique in the proof of Theorem 3.1.
Lemma 2.5 (see [39, Theorem 1, page 300]).
Let with
, and let
with
. If
for
, then the solution of the differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ17_HTML.gif)
with is analytic in
and satisfies
.
Lemma 2.6 (see [1, Lemma 2.2d, page 24]).
Let with
, and let
be analytic in
with
and
. If
is not subordinate to
, then there exists points
and
, for which
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ18_HTML.gif)
A function defined on
is a subordination chain (or Löwner chain) if
is analytic and univalent in
for all
,
is continuously differentiable on
for all
, and
for
.
Lemma 2.7 (see [3, Theorem 7, page 822]).
Let ,
, and set
. If
is a subordination chain and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ19_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ20_HTML.gif)
Furthermore, if has a univalent solution
, then
is the best subordinant.
Lemma 2.8 (see [3, Lemma B, page 822]).
The function , with
and
, is a subordination chain if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ21_HTML.gif)
3. Main Results
We first prove the following subordination theorem involving the operator defined by (1.10).
Theorem 3.1.
Let . For
,
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ22_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ23_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ24_HTML.gif)
Then the subordination condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ25_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ26_HTML.gif)
Moreover, the function is the best dominant.
Proof.
Let us define the functions and
, respectively, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ27_HTML.gif)
We first show that if the function is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ28_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ29_HTML.gif)
Logarithmic differentiation of both sides of the second equation in (3.6) and using (1.11) for yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ30_HTML.gif)
Now, differentiating both sides of (3.9) results in the following relationship:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ31_HTML.gif)
We also note from (3.2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ32_HTML.gif)
and, by using Lemma 2.5, we conclude that differential equation (3.10) has a solution with
. Let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ33_HTML.gif)
where is given by (3.3). From (3.2), (3.10), and (3.12), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ34_HTML.gif)
In order to use Lemma 2.4, we now proceed to show that for all real
and
. Indeed, from (3.12),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ35_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ36_HTML.gif)
For given by (3.3), we can prove easily that the expression
given by (3.15) is positive or equal to zero. Hence, from (3.14), we see that
for all real
and
. Thus, by using Lemma 2.4, we conclude that
for all
. That is,
defined by (3.6) is convex in
. Next, we prove that subordination condition (3.4) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ37_HTML.gif)
for the functions and
defined by (3.6). Without loss of generality, we also can assume that
is analytic and univalent on
and
for
. For this purpose, we consider the function
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ38_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ39_HTML.gif)
This shows that the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ40_HTML.gif)
satisfies the condition for all
. Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ41_HTML.gif)
Therefore, by virtue of Lemma 2.8, is a subordination chain. We observe from the definition of a subordination chain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ42_HTML.gif)
Now suppose that is not subordinate to
; then, by Lemma 2.6, there exist points
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ43_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ44_HTML.gif)
by virtue of subordination condition (3.4). This contradicts the above observation that . Therefore, subordination condition (3.4) must imply the subordination given by (3.16). Considering
, we see that the function
is the best dominant. This evidently completes the proof of Theorem 3.1.
We next prove a dual result to Theorem 3.1, in the sense that subordinations are replaced by superordinations.
Theorem 3.2.
Let . For
,
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ45_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ46_HTML.gif)
where is given by (3.3). Further, suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ47_HTML.gif)
is univalent in and
. Then the superordination
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ48_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ49_HTML.gif)
Moreover, the function is the best subordinant.
Proof.
The first part of the proof is similar to that of Theorem 3.1 and so we will use the same notation as in the proof of Theorem 3.1.
Now let us define the functions and
, respectively, by (3.6). We first note that if the function
is defined by (3.7), then (3.9) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ50_HTML.gif)
After a simple calculation, (3.29) yields the relationship
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ51_HTML.gif)
Then by using the same method as in the proof of Theorem 3.1, we can prove that for all
. That is,
defined by (3.6) is convex (univalent) in
. Next, we prove that the subordination condition (3.27) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ52_HTML.gif)
for the functions and
defined by (3.6). Now considering the function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ53_HTML.gif)
we can prove easily that is a subordination chain as in the proof of Theorem 3.1. Therefore according to Lemma 2.7, we conclude that superordination condition (3.27) must imply the superordination given by (3.31). Furthermore, since the differential equation (3.29) has the univalent solution
, it is the best subordinant of the given differential superordination. This completes the proof of Theorem 3.2.
Combining Theorems 3.1 and 3.2, we obtain the following sandwich-type theorem.
Theorem 3.3.
Let . For
,
,
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ54_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ55_HTML.gif)
where is given by (3.2). Further, suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ56_HTML.gif)
is univalent in and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ57_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ58_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
The assumption of Theorem 3.3 that the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ59_HTML.gif)
need to be univalent in may be replaced by another condition in the following result.
Corollary 3.4.
Let . For
,
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ60_HTML.gif)
and ,
be as in (3.33). Suppose that condition (3.34) is satisfied and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ61_HTML.gif)
where is given by (3.3). Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ62_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ63_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
Proof.
In order to prove Corollary 3.4, we have to show that condition (3.40) implies the univalence of and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ64_HTML.gif)
Since given by (3.3) in Theorem 3.1 satisfies the inequality
, condition (3.40) means that
is a close-to-convex function in
(see [40]) and hence
is univalent in
. Furthermore, by using the same techniques as in the proof of Theorem 3.1, we can prove the convexity (univalence) of
and so the details may be omitted. Therefore, from Theorem 3.3, we obtain Corollary 3.4.
By taking ,
,
,
, and
in Theorem 3.3, we have the following result.
Corollary 3.5.
Let . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ65_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ66_HTML.gif)
and is univalent in
and
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ67_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ68_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
Next consider the generalized Libera integral operator defined by (cf. [37, 41–43])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ69_HTML.gif)
For the choice , with
, (3.48) reduces to the well-known Bernardi integral operator [41]. The following is a sandwich-type result involving the generalized Libera integral operator
.
Theorem 3.6.
Let . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ70_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ71_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ72_HTML.gif)
If is univalent in
and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ73_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ74_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
Proof.
Let us define the functions and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ75_HTML.gif)
respectively. From the definition of the integral operator given by (3.48), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ76_HTML.gif)
Then, from (3.49) and (3.55),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ77_HTML.gif)
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ78_HTML.gif)
and differentiating both sides of (3.51) result in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ79_HTML.gif)
The remaining part of the proof is similar to that of Theorem 3.3 (a combined proof of Theorems 3.1 and 3.2) and is therefore omitted.
By using the same methods as in the proof of Corollary 3.4, the following result is obtained.
Corollary 3.7.
Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ80_HTML.gif)
Suppose that condition (3.50) is satisfied and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ81_HTML.gif)
where is given by (3.51). Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ82_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ83_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
Taking ,
,
, and
in Corollary 3.7, we have the following result.
Corollary 3.8.
Let . Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ84_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ85_HTML.gif)
where is given by (3.51), and
is univalent in
and
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ86_HTML.gif)
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F486595/MediaObjects/13660_2010_Article_2344_Equ87_HTML.gif)
Moreover, the functions and
are the best subordinant and the best dominant, respectively.
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Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2010-0017111) and grants from Universiti Sains Malaysia and University of Delhi. The authors are thankful to the referees for their useful comments.
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Cho, N.E., Kwon, O.S., Ali, R.M. et al. Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator. J Inequal Appl 2011, 486595 (2011). https://doi.org/10.1155/2011/486595
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DOI: https://doi.org/10.1155/2011/486595