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Journal of Complex Analysis
Volume 2014 (2014), Article ID 187482, 12 pages
Subordination Properties of Multivalent Functions Defined by Generalized Multiplier Transformation
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.
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
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.  and Sivaprasad Kumar et al. .(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  and Orhan and Kiziltunç  and also, for and , it yields the differential operator introduced by Sălăgean .(4)As a special case of this operator for and , it reduces the generalized Sălăgean operator studied by Al-Oboudi  and also earlier, for , it gives the operator investigated by Uralegaddi and Somanatha .
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.  and Patel and Mishra , 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.
To prove our results, we will need the following lemmas.
We denote by the class of functions given by (13) which are analytic in and satisfy the following inequality:
Lemma 4 (see ). For , The result is the best possible.
For any complex numbers , , and , the Gaussian hypergeometric function is defined by
Lemma 5 (see ). For any complex numbers , , (), one has
Lemma 6 (see ). 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 ). 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 ). Let be a real number, , and . Let be analytic in and where If is analytic in and satisfies the subordination relation then for .
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
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
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.
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
which is a positive measure on the closed interval , we get
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
Theorem 14. If and is given by (70), satifies the subordination condition: then where The result is the best possible.
4. Inclusion Relationship for the Class
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
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 .
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 .
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
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.
The authors thank the referees for their insightful suggestions.
- S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.