Research Article | Open Access

# Subordination Properties of Multivalent Functions Defined by Generalized Multiplier Transformation

**Academic Editor:**Janne Heittokangas

#### Abstract

The main object of the present paper is to investigate several interesting subordination properties and a sharp inclusion relationship for certain subclass of multivalent analytic functions, which are defined here by the generalized multiplier transformation. Relevant connections of the results which are presented in this paper with various known results are also considered.

#### 1. Introduction and Definitions

Let denote the class of analytic functions in the open unit disk
If and are in , we say that the function is said to be *subordinate* to or (equivalently) is said to be *superordinate* to , written symbolically as
if there exists a *Schwarz function*â€‰â€‰ analytic in , with and , for all , such that
In particular, if the function is univalent in , then we have the following equivalence (cf. [1, 2]):

Let be the subclass of consisting of functions defined by
which are analytic and *-valent* in the open unit disk . We note that .

For a function given by (5) and defined by
the *Hadamard product* (or *convolution*) of and is given by

Recently, CÄƒtaÅŸ [3] defined the *generalized multiplier transformation*â€‰â€‰ on by the following infinite series:
We note that
It is easily verified from (8) that

The generalized multiplier transformation reduces several familiar operators by specializing the parameters , and .(1)For the choice of , the operator defined by (8) reduces the operator , studied by Srivastava et al. [4] and Sivaprasad Kumar et al. [5].(2)By taking , the generalized multiplier transformation yields the operator , which was investigated by Cho et al. [6, 7].(3)For and , the operator reduces the differential operator studied by Kamali and Orhan [8] and Orhan and KiziltunÃ§ [9] and also, for and , it yields the differential operator introduced by SÄƒlÄƒgean [10].(4)As a special case of this operator for and , it reduces the generalized SÄƒlÄƒgean operator studied by Al-Oboudi [11] and also earlier, for , it gives the operator investigated by Uralegaddi and Somanatha [12].

Now, we introduce a new subclass of functions in , by making use of the generalized multiplier transformation as follows.

*Definition 1. *Let , , , , , be arbitrary fixed real numbers such that , , , , and . A function is said to be in the class , if it satisfies the following subordination condition:

In particular, for and , we write , where

Motivated by the recent work of BulboacÄƒ et al. [13] and Patel and Mishra [14], we investigate the subordination properties of the generalized multiplier transformation defined by (8) and obtain a sharp inclusion relationship for the multivalent analytic function class . We also derive a number of sufficient conditions for functions belonging to the subclass which satisfy certain subordination properties. Relevant connections of the results presented in this paper with earlier sequels are also pointed out.

#### 2. Preliminaries

To prove our results, we will need the following lemmas.

Lemma 2 (see [1, 2]). *Let a function be analytic and convex (univalent) in , with . Suppose also that the function given by
**
is analytic in . If
**
then
**
where is the best dominant of (14).*

We denote by the class of functions given by (13) which are analytic in and satisfy the following inequality:

Lemma 3 (see [15]). *Let the function given by (13) be in the class . Then
*

Lemma 4 (see [16]). *For ,
**
The result is the best possible.*

For any complex numbers , , and , the *Gaussian hypergeometric function* is defined by

Lemma 5 (see [17]). *For any complex numbers , , (), one has
*

Lemma 6 (see [18]). *If , , and the complex number is constrained by , then the following differential equation:
**
has a univalent solution in given by
**
If the function given by (13) is analytic in and satisfies the following subordination:
**
then
**
and is the best dominant of (23).*

Lemma 7 (see [19]). *Let be a positive measure on the interval . Let be a complex-valued function defined on such that is analytic in for each and is -integrable on for each . In addition, suppose that , is real, and
**
If the function is defined by
**
then
*

Lemma 8 (see [20]). *Let be a real number, , and . Let be analytic in and
**
where
**
If is analytic in and satisfies the subordination relation
**
then for .*

Lemma 9 (see [2]). *Suppose that the function satisfies the following condition:
**
for all and and for all . If the function of the form (13) is analytic in and
**
then
*

#### 3. Subordination Properties of

Unless otherwise mentioned, we assume throughout this paper that

Theorem 10. *Let and , . If the functions satisfy the following subordination condition:
**
then
**
where and
**
The result is the best possible when .*

*Proof. *Let the functions , , satisfy the subordination condition (35). Then, by setting
we have
By making use of (10) and (38), we obtain
Now, if we let , then by using (40) and the fact that
a simple computation shows that
where
Since , , it follows from Lemma 4 that
and the bound is the best possible. Hence, by using Lemma 3 in (43), we deduce that
where is given by (37).

When , we consider the functions â€‰â€‰ which satisfy the hypothesis (35) and are given by
Since
it follows from (43) that
Therefore
which evidently completes our proof of Theorem 10.

By setting , , and , , in Theorem 10, we have the following corollary.

Corollary 11. *If the functions satisfy the following subordination condition:
**
then
**
where .*

In Theorem 12, we have determined the sufficient condition for the functions to be a member of the class .

Theorem 12. *If satisfy the following subordination condition:
**
then
**
where
**
The result is the best possible.*

*Proof. *Let
Then, the function is of the form (13). Differentiating (55) with respect to and using the identity (10), we obtain
By using (52), (55), and (56), we get
Now, by applying Lemma 2, we have
By using Lemma 5, we get
Now, we will show that
We have
and setting
which is a positive measure on the closed interval , we get
so that
As in (64), we obtain the assertion (60). Now, by using (59) and (60), we get
where is given by (54).

To show that the estimate (54) is the best possible, we consider the function defined by
For the above function, we find that
as , and the proof of the Theorem 12 is completed.

In its special case when , and , Theorem 12 yields the following corollary.

Corollary 13. *If satisfy the following condition:
**
then
**
The result is the best possible.*

For a function , the integral operator

Also, it is easily verified from (70) that

In the next Theorem 14, by using the integral operator defined in (70), we established the sufficient condition for the functions belongs to .

Theorem 14. *If and is given by (70), satifies the subordination condition:
**
then
**
where
**
The result is the best possible.*

*Proof. *Let
Then by using the hypothesis (72) together with (71) and (75), we obtain
The remaining part of the proof of Theorem 14 is similar to that of Theorem 12 and hence we omit the details.

#### 4. Inclusion Relationship for the Class

Theorem 15. *If and
**
then
**
where
**
and is the best dominant of (78). If, in addition to (77),
**
then
**
where
**
The bound on is the best possible.*

*Proof. *Let . Define the function by
and . Then is single-valued and analytic function in . By logarithmic differentiation in (83), it follows that the function given by
is analytic in and . Using the identity (10) in (84) and logarithmic differentiation of the resulting equation yields the following:
Hence, by using Lemma 6 with and , we find that
where is the best dominant of (86) and is given by (79). Since
by (86), we have . Now, (84) shows that is starlike (univalent) in . Thus, it is not possible that vanishes on if . So, we conclude that , and, therefore, is analytic in . Hence, (86) implies that
This proves the assertion (78) of Theorem 15.

In order to establish (81), we have to find the greatest lower bound of such that
By (86), we have to show that
To prove (90), we need to show that
From (79), we see that, for ,
where
Since , by using Lemma 5, we get following:
Since
implies that , by using Lemma 5, we find from (94) that
where
which is positive measure on . For , it may be noted that and is real for and . Hence, by using Lemma 7, we have
We note that . Thus, by using (90) and (94), we have
when . Further by taking â€‰for the case and using (78), we get (81). The result is the best possible as the function is the best dominant of (78). This completes the proof of Theorem 15.

In the following section, we obtain the sufficient condition for the function to be a member of the class .

#### 5. Sufficient Conditions for the Class

Theorem 16. *If satisfy the following subordination condition:
**
where
**
then .*

*Proof. *Let
Then, the function is of the form (13) and is analytic in . From Theorem 12 with and , we have
which is equivalent to
If we set
Then, by using the identity (10) followed by (102), we obtain
In view of (106), the hypothesis (100) can be written as follows:
We need to show that (107) yields
Suppose that this is false. Since , there exists a point such that for some . Therefore, in order to show that (108), it is sufficient to obtain the contradiction from the inequality
If we let , then, by using (104) and the triangle inequality, we obtain that
If we let
then (109) holds true if , for any . Since , the inequality holds true if the discriminant ; that is,
which is equivalent to
After a simple computation, by using (104), we obtain the inequality