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Journal of Complex Analysis
Volume 2013 (2013), Article ID 502363, 3 pages
Some Results of Univalent and Starlike Integral Operator
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.
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 ). 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 ). Let the functions and be analytic in , and let map onto a starlike region. Suppose also that Then, for all .
Lemma 3 (see ). Let Then, for and , for all .
Lemma 4 (see ). 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 .
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
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.
The work presented here was supported by LRGS/TD/2011/UKM/ICT/03/02.
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