Inclusion Properties of Certain Subclasses of p-Valent Functions Associated with the Integral Operator

Tamer M. Seoudy Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt Correspondence should be addressed to Tamer M. Seoudy; tms00@fayoum.edu.eg Received 2 May 2014; Accepted 18 May 2014; Published 29 May 2014 Academic Editor: Ming-Sheng Liu Copyright © 2014 Tamer M. Seoudy. 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.

We note that (i) the one-parameter family of integral operator   ,1 =    was defined by Jung et al. [5] and studied by Aouf [7] and Gao et al. [8].
We have the following known subclasses S  (, ) and C  (, ) of the class A() for 0 ≤ ,  < , and  ≥ 2 which are defined by Next, by using the integral operator   , , we introduce the following classes of analytic functions for 0 ≤  <  and  ≥ 2: We also note that In particular, we set S  (1, ; ) = S  (; ) and C  (1, ; ) = C  (; ).
The following lemma will be required in our investigation.
In this paper, we obtain several inclusion properties of the classes S  (, ; ) and C  (, ; ) associated with the operator   , .
where ℎ  is analytic in U with ℎ  (0) = 1,  = 1, 2. Using the identity ( 6) in ( 14) and differentiating the resulting equation with respect to , we obtain This implies that We form the functional Ψ(, V) by choosing  = ℎ  () and V = ℎ   (): Clearly, the first two conditions of Lemma 1 are satisfied.Now, we verify condition (iii) as follows: Therefore applying Lemma 1, ℎ  ∈ P ( = 1, 2) and consequently ℎ ∈ P  for  ∈ U.This completes the proof of Theorem 4.
Theorem 5.One has Proof.Applying (10) and Theorem 4, we observe that which evidently proves Theorem 5.
Next, we derive an inclusion property for the subclass C  (; ) involving  , (), which is given by the following theorem.