Subordination Properties of Multivalent Functions Defined by Generalized Multiplier Transformation

M. P. Jeyaraman and T. K. Suresh 1 Department 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 Correspondence should be addressed to M. P. Jeyaraman; jeyaraman mp@yahoo.co.in 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. Themain 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 generalizedmultiplier transformation. Relevant connections of the results which are presented in this paper with various known results are also considered.


Introduction and Definitions
The generalized multiplier transformation    (, ) reduces several familiar operators by specializing the parameters , , , and .
(4) As a special case of this operator    (, ) for  = 0 and  = 1, it reduces the generalized Sȃlȃgean operator    studied by Al-Oboudi [11] and also earlier, for  =  =  = 1, it gives the operator   investigated by Uralegaddi and Somanatha [12].Now, we introduce a new subclass of functions in A(), by making use of the generalized multiplier transformation    (, ) as follows.
Definition 1.Let  ∈ N 0 , , , , ,  be arbitrary fixed real numbers such that −1 ≤  <  ≤ 1, 0 ≤  < ,  > 0,  ∈ N, and  ≥ 0. A function  ∈ A() is said to be in the class R   (, , ; , ), if it satisfies the following subordination condition: In particular, for  = 1 and  = −1, we write R   (, , ; 1, −1) = R   (, , ), where Motivated by the recent work of Bulboacȃ 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 R   (, , ; , ).We also derive a number of sufficient conditions for functions belonging to the subclass R   (, , ) which satisfy certain subordination properties.Relevant connections of the results presented in this paper with earlier sequels are also pointed out.

Preliminaries
To prove our results, we will need the following lemmas.
Lemma 2 (see [1,2]).Let a function ℎ be analytic and convex (univalent) in U, with ℎ(0) = 1.Suppose also that the function  given by then where  is the best dominant of (14).
We denote by () the class of functions  given by (13) which are analytic in U and satisfy the following inequality: Lemma 3 (see [15]).Let the function  given by (13) be in the class ().Then Lemma 4 (see [16]).
The result is the best possible.
In Theorem 12, we have determined the sufficient condition for the functions    (, )()/  to be a member of the class ().
Theorem 12.If  ∈ A() 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 inf {Re  () : || < 1} =  (−1) .
We have and setting which is a positive measure on the closed interval [0, 1], we get As  → 1 − in (64), we obtain the assertion (60).Now, by using ( 59) and (60), we get where  is given by (54). To as  → −1, and the proof of the Theorem 12 is completed.

Corollary 13. If 𝑓 ∈ A(𝑝) satisfy the following condition:
then The result is the best possible.(71) In the next Theorem 14, by using the integral operator defined in (70), we established the sufficient condition for the functions    (, ) , ()/  belongs to ( 0 ). 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.

Inclusion Relationship for the
where and () is the best dominant of (78).If, in addition to (77), where The bound on  is the best possible.
In order to establish (81), we have to find the greatest lower bound of  (0 <  < 1) such that By (86), we have to show that To prove (90), we need to show that From (79), we see that, for  ̸ = 0, implies that  >  > 0, by using Lemma 5, we find from (94) that where We note that (−1) ̸ = 0. Thus, by using ( 90) and (94), we have In the following section, we obtain the sufficient condition for the function  to be a member of the class R   (, , ).

Sufficient
Then, the function  is of the form (13) and is analytic in U.
From Theorem 12 with  =  1 and  = 0, we have which is equivalent to If we set Then, by using the identity (10) After a simple computation, by using (104), we obtain the inequality