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

Abstract

The purpose of the present paper is to introduce two subclasses of -valent functions by using the integral operator and to investigate various properties for these subclasses.

1. Introduction

Let denote the class of functions of the following form: which are analytic and -valent in the open unit disc . Let be the class of functions analytic in satisfying and The class was introduced by Aouf [1] and we note the following:(i)the class was introduced by Padmanabhan and Parvatham [2];(ii)the class was introduced by Pinchuk [3];(iii) is the class of functions with positive real part greater than ;(iv) is the class of functions with positive real part greater than ;(v) is the class of functions with positive real part.

From (1), we have if and only if there exists such that It is known that [4] the class is a convex set.

Motivated essentially by Jung et al. [5], Liu and Owa [6] introduced the integral operator as follows: For given by (1) and then from (4), we deduce that It is easily verified from (5) that (see [6]) We note that (i) the one-parameter family of integral operator was defined by Jung et al. [5] and studied by Aouf [7] and Gao et al. [8].

(ii) Consider where the operator is the generalized Bernardi-Libera-Livingston integral operator (see [9]).

We have the following known subclasses and of the class for , , and which are defined by

Next, by using the integral operator , we introduce the following classes of analytic functions for and : We also note that In particular, we set and .

The following lemma will be required in our investigation.

Lemma 1 (see [10]). Let and and let be a complex-valued function satisfying the following conditions:(i) is continuous in a domain ;(ii) and ;(iii) whenever and .
If is analytic in such that and for , then in .

Lemma 2 (see [11]). Let be analytic in with and ,  . Then, for and , where is given by and this radius is the best possible.

Lemma 3 (see [12]). Let be convex and let be starlike in . Then, for analytic in with , is contained in the convex hull of .

In this paper, we obtain several inclusion properties of the classes and associated with the operator .

2. Main Results

Unless otherwise mentioned, we assume throughout this paper that , , , , and .

Theorem 4. One has

Proof. We begin by setting where is analytic in with , . Using the identity (6) in (14) and differentiating the resulting equation with respect to , we obtain This implies that We form the functional by choosing and :
Clearly, the first two conditions of Lemma 1 are satisfied. Now, we verify condition (iii) as follows: Therefore applying Lemma 1, and consequently for . 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.

Theorem 6. If , then , where the generalized Libera integral operator is defined by (7).

Proof. Let and set where is analytic in with . From (21), we have Then, by using (21) and (22), we obtain Taking the logarithmic differentiation on both sides of (23) with respect to and multiplying by , we have This implies that We form the functional by choosing and : Then clearly satisfies all the properties of Lemma 1. Hence, and consequently for , which implies that .

Next, we derive an inclusion property for the subclass involving , which is given by the following theorem.

Theorem 7. If , then , where is defined by (7).

Proof. By applying Theorem 6, it follows that
which proves Theorem 7.

Theorem 8. If , for , then for where , with and . This radius is the best possible.

Proof. Let for and let where is analytic in with and for . Using the identity (6) in (29) and differentiating the resulting equation with respect to , we obtain where for . Applying Lemma 2 with and , we get where is given by (28). This completes the proof of Theorem 8.

Theorem 9. Let be a convex function and . Then , where .

Proof. Let . Then Also, . Therefore, . By logarithmic differentiation of (32) and after some simplification, we obtain where is analytic in and . From Lemma 3, we can see that is contained in the convex hull of . Since is analytic in and then lies in ; this implies that .

Remark 10. Putting in the above results, we obtain corresponding results for the operator .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.