Journal of Complex Analysis

Volume 2013 (2013), Article ID 502363, 3 pages

http://dx.doi.org/10.1155/2013/502363

Research Article

Some Results of Univalent and Starlike Integral Operator

E. A. Eljamal,¹ M. Darus,¹ and D. Breaz²

¹School of Mathematical Sciences, Faculty of Science and Technology, The National University of Malaysia, Selangor “D. Ehsan,” 43600 Bangi, Malaysia

²Department of Mathematics-Informatics, Faculty of Science, 1 Decembrie 1918 University of Alba Iulia, 510009 Alba Iulia, Romania

Received 14 October 2012; Accepted 24 October 2012

Academic Editor: Nikolai Tarkhanov

Copyright © 2013 E. A. Eljamal et al. 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 main objective of the present paper is to study the mapping properties of functions belonging to certain classes under a family of univalent and starlike integral operator. Relationships of these classes are also pointed out.

1. Introduction and Definitions

Let denotes the class of functions normalized by

which are analytic in the open unit disk

Also let denotes the class of all functions in which are univalent in . Then a function is said to be starlike in if and only if

We denote by the class of all functions in which are starlike in . A function is said to be starlike of order in if and only if

for some . We denote by the class of all functions in which are starlike of order in . Clearly, we have .

With a view to introducing an interesting family of analytic functions, we should recall the concept of subordination between analytic functions. Given two functions and , which are analytic in , the function is said to be subordinate to if there exists a function , analytic in with and    such that , , and symbolically written as the following:

It is known that    and .

Definition 1 (see [1]). For , a function , analytic in with , is said to belong to the class if

To prove our main result, we need the following.

Lemma 2 (see [2]). Let the functions and be analytic in , and let map onto a starlike region. Suppose also that

Then,

for all .

Lemma 3 (see [3]). Let

Then, for and ,

for all .

Lemma 4 (see [4]). Let the functions and be analytic in with

and let be a real number. Suppose also that maps onto a region which is starlike with respect to the origin. Then,

2. Main Result

We begin by introducing a new integral operator

where and .

Bear in mind, there are various types of integral operators studied by many different authors such as [5–9], few to mention, that motivate us to come out with the abovementioned integral operators.

Now let us begin with our first result relating to the integral operator of (13).

Theorem 5. Let the functions and be in the class . Then, the function defined by (13) is also in the class .

Proof. By logarithmic differentiation, we find from (13) that

where

Clearly, we have , and satisfies the starlikeness condition of Lemma 4.

Next, let , where is analytic function in , and . From (15), it is easily seen that

hence by Lemma 4, we obtain

that is

which evidently proves Theorem 5.

Theorem 6. Let the functions and be in the class . Then, is in the class .

Proof. Since

we find from Definition 1 that

By logarithmic differentiation, we find from (13) that

where

Next, let , and from (15), it is easily seen that

Now rewrite the equality in (23) in the form

so that by (20) and Lemma 3, we have

It is easily seen that and satisfy conditions of Lemma 2. It follows from (21), (25), and Lemma 2 that

which evidently proves Theorem 6.

Acknowledgment

The work presented here was supported by LRGS/TD/2011/UKM/ICT/03/02.

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