Journal of Complex Analysis

Volume 2014 (2014), Article ID 187482, 12 pages

http://dx.doi.org/10.1155/2014/187482

Research Article

Subordination Properties of Multivalent Functions Defined by Generalized Multiplier Transformation

M. P. Jeyaraman¹ and T. K. Suresh²

¹Department of Mathematics, L. N. Government College, Ponneri, Chennai, Tamil Nadu 601 204, India

²Department of Mathematics, Easwari Engineering College, Chennai, Tamil Nadu 600 089, India

Received 31 August 2013; Accepted 11 November 2013; Published 25 February 2014

Academic Editor: Janne Heittokangas

Copyright © 2014 M. P. Jeyaraman and T. K. Suresh. 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.

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   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, Cătaş [3] defined the generalized multiplier transformation   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 Kiziltunç [9] and also, for and , it yields the differential operator introduced by Sălăgean [10].(4)As a special case of this operator for and , it reduces the generalized Sălăgean 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 Bulboacă 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    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  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

which yields . Therefore , which contradicts (107). It follows that , and .

Theorem 17. Let and if such that , , satisfies the following differential subordination:

where the powers are understood as the principle value, and

then .

Proof . If , then the condition (115) is equivalent to

The above equation (117) implies that .

If we consider and suppose that

Choosing the principal value in (118), we note that is of the form (13) and is analytic in . Differentiating (118) with respect to , we obtain

which, in view of Lemma 2 with , yields

Also, with the aid of (118), (115) can be written as follows:

where is given by (105). Therefore, by Lemma 8, we find that

that is

which completes the proof of Theorem 17

By taking in Theorem 17, we get the following corollary due to Patel et al. [21].

Corollary 18. Let , , , and . If such that   , and satisfies the following subordination condition:

where the powers are understood as the principal value and is given by (116), then .

Remark 19. Taking and in Theorem 17, we get the result of Liu [20, Theorem  2.2, with ].

Remark 20. Putting and in Theorem 17, we obtain the result of Liu [20, Corollary  2.1, with ].

Remark 21. Putting and , in Theorem 17, we get the result of Mocanu and Oros [22, Corollary  2.2, with ].

Remark 22. Taking , and in Theorem 17, we obtain the result of Mocanu [23, with ].

Taking and , in Theorem 17, we obtain the following corollary.

Corollary 23. Let , , and . If such that satisfies the inequality:

where the powers are the principal value ones, then .

Remark 24. Taking in Corollary 23, we obtain the result of Liu [20, Corollary  2.2, with ].

Remark 25. Putting in Corollary 23, we get the result of Mocanu and Oros [22, Corollary  2.4, with ].

In Theorem 26, we obtain the necessary condition for functions belonging to class .

Theorem 26. Let (). If the functions   , then the function , satisfies the following inequality:

provided that

Proof. Let . By using Lemma 4 and (10), we obtain

Then, by using Lemma 2 with , , and , we have

Again, from (129) and Corollary 13, we obtain

Now, if we let    then is of the form (13) and is analytic in . A simple computation shows that

where .

Thus, by using (128), (131) can be written as follows:

For all real and , we have

where we used (127) and (130). Thus, by Lemma 9, we get

which completes the proof of Theorem 26.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the referees for their insightful suggestions.

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