Univalency and Convexity Conditions for a General Integral Operator

For analytic functions and in the open unit disc , a new general integral operator is introduced. The main objective of this paper is to obtain univalence condition and order of convexity for this general integral operator.


Introduction and Preliminaries
Let A be the class of all functions of the form which are analytic in the open unit disk Also let S denote the subclass of A consisting of functions  which are univalent in U.
A function  ∈ A is said to be starlike of order  (0 ≤  < 1) if it satisfies the inequality Re {   ()  () } > , for all  ∈ U. We say that  is in the class S * () for such functions.
A function  ∈ A is said to be convex of order  (0 ≤  < 1) if it satisfies the inequality Re {1 +   ()   () } > , for all  ∈ U. We say that  is in the class K() for such functions.
We note that  ∈ K() if and only if   ∈ S * ().
A function  ∈ A belongs to the class R() (0 ≤  < 1) if it satisfies the inequality Re {  ()} > , for all  ∈ U.
Remark 1.This family is a comprehensive class of analytic functions that contains other new classes of analytic univalent functions as well as some very well-known ones.For example, (i) for  = 1, we have the class (ii) for  = 0, we have the class (iii) for  = 2, the class introduced by Frasin and Darus [2].

Chinese Journal of Mathematics
In this paper, we introduce a new general integral operator defined by where  ∈ U,   ,   ∈ A, and   ∈ C for all  = 1, . . ., .
The following results will be required in our investigation.
General Schwarz Lemma (see [4]).Let the function  be regular in the disk U  = { ∈ C : || < }, with |()| <  for fixed .If  has one zero with multiplicity order bigger than  for  = 0, then The equality can hold only if () =   (/  )   , where  is constant.
Theorem A (see [5]).If the function  is regular in the unit disk U, () =  +  2  2 + ⋅ ⋅ ⋅ and for all  ∈ U, then the function  is univalent in U.
then  is univalent in U.

Main Results
Theorem 3. Let   ,   ∈ A, where   satisfies the condition for all  = 1, . . ., , then the integral operator  , (, ) defined by (10) is in the univalent function class S.