Sufficient conditions for new integral transformation

By using the integral transformation for m eromorphic functions, an integral transformation on the class A of analytic functions in the unit disk is defined. Some sufficient conditions for this transformation to be in the some known subclasses are derived. Furthermore, new function on the class of meromorphic functions in the punctured open unit disk is introduced. Finally, starlikeness conditions for this function are pointed out.


INTRODUCTION
Let Σ denote the class of functions of the form

U z C z =∈ <
We say that a function f ∈ Σ is the meromorphic starlike of order ( ) A function f ∈ Σ is the meromorphic convex function of order ( ) C α and UCV denote the subclasses of A consisting of functions which are, respectively, starlike, convex and close-to-convex of order ( ) and uniformly convex function.Thus, we have '( ) ( ) : and , 0 1; , ( ) In addition, let ( ) N β be the subclass of A consisting of functions f which satisfy the inequality: ''( ) 1 , , 1. '( ) This class was introduced and studied by (Owa and Srivastava, 2002).
The study of integral operators finds an important place in the field of Geometric Function Theory not only in the past, even recently.The Alexander transformation was introduced by Alexander (1915).The surprisingly close analytic connection between the well-known two subclasses of the class of univalent functions, namely the class of starlike functions and the class of convex functions was first discovered by Alexander by using his transformation.Later Libera (1965) introduced an integral operator which generalized by Bernardi (1969).Recently, new frontiers of integral operators are designed to stimulate interest among the young researchers and new readers to the area of geometric function theory (cf., e.g., [1,[4][5][7][8][9][10][11]).
For the sake of simplicity, we will write , , , ( ).
In order to derive our main results, we have to recall here the following lemmas.Lemma 1.1.6(Cho, Owa, 2001 where Main results

Starlikeness of the function ( ) z Θ
In this section we place conditions for the function ( ) z Θ which is defined in (1.1.6),to be in the class ( ).
( ) then the function ( ) z Θ given by (1.1.6) Taking the real part of both terms of (2.2.2), we get But by the hypothesis, ( ) .  ( ) By multiplying the above expression with , z we obtain ( ) ( ) Taking the real part of both terms of (2.2.4), we get    Since the proof is similar to the proof of theorem 2.2.3, it will be omitted.

CONCLUSION
Integral operators have been of interests to many in recent years and will continue as far as research is concerned.Many new results and properties are obtained with different types and styles of operators being defined.

Definition 1 . 1 . 1 .
(Mohammed, Darus, 2010) Let i z on the classes ( ), ( ), this section, we investigate the conditions under which the integral operator ( ) F z defined in(1.1.3)to be in the classes is a starlike function.