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Journal of Complex Analysis
Volume 2013 (2013), Article ID 785015, 3 pages
http://dx.doi.org/10.1155/2013/785015
Research Article

Notes on Certain Multivalent Analytic Functions Associated with a Linear Operator

Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China

Received 1 October 2012; Accepted 15 November 2012

Academic Editor: Bao Qin Li

Copyright © 2013 Jin-Lin Liu. 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 purpose of the present paper is to investigate some properties of multivalent analytic functions.

1. Introduction

Let denote the class of the functions of the form which are analytic in the open unit disk . Also let the Hadamard product (or convolution) of two functions be given by

In [1], Liu introduced the following generalized Srivastava-Attiya operator : where It is not difficult to see from (5) and (6) that When , the operator is the well-known Srivastava-Attiya operator [2]. The generalized Srivastava-Attiya operator has been studied by several authors (see [15]).

In this investigation, we focus on certain inequalities consisting of the following differential operator: Our results generalize the recent results obtained by Irmak et al. [6].

In order to prove our main results, we need the following lemmas.

Lemma 1 (see [7]). Let and suppose that the function satisfies for all , , and . If is analytic in and for all , then .

Lemma 2 (see [7]). Let and suppose that the function satisfies for all , , and . If is analytic in and for all , then .

2. Main Results

We now state and then prove each of our main results given by Theorems 3 and 4 below.

Theorem 3. Let with for all , and also let and . If where , then

Proof. Put Then the function is analytic in with . A simple computation shows that
Now letting we obtain that for all . Further, for any , , and , since , we also have which shows that . Therefore, according to Lemma 1, we obtain . This completes the proof of Theorem 3.

Theorem 4. Let with for , and also let and . If where then

Proof. Suppose that Then is analytic in . It is easily seen from (18) that Further, since it leads to for . Also, for any , and , we have that is, . Finally, by Lemma 2, we obtain that . The proof of Theorem 4 is completed.

Remark 5. For and , Theorems 3 and 4 reduce to the results obtained by Irmak et al. [6].

References

  1. J.-L. Liu, “Subordinations for certain multivalent analytic functions associated with the generalized Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 19, pp. 893–901, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. M. Srivastava and A. A. Attiya, “An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination,” Integral Transforms and Special Functions, vol. 18, pp. 207–216, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. N. E. Cho, I. H. Kim, and H. M. Srivastava, “Sandwich-type theorems for multivalent functions associated with the Srivastava-Attiya operator,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 918–928, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Murugusundaramoorthy, “Subordination results for spiral-like functions associated with the Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 23, no. 2, pp. 97–103, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Z.-G. Wang, Z.-H. Liu, and Y. Sun, “Some properties of the generalized Srivastava-Attiya operator,” Integral Transforms and Special Functions, vol. 23, no. 3, pp. 223–236, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Irmak, T. Bulboacă, and N. Tuneski, “Some relations between certain classes consisting of α-convex type and Bazilević type functions,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2010–2014, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet