Abstract
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.
1. Introduction and Preliminaries
Let be the class of all functions of the form which are analytic in the open unit disk Also let denote the subclass of consisting of functions which are univalent in .
A function is said to be starlike of order if it satisfies the inequality for all . We say that is in the class for such functions.
A function is said to be convex of order if it satisfies the inequality for all . We say that is in the class for such functions.
We note that if and only if .
A function belongs to the class if it satisfies the inequality for all .
The family which contains the functions that satisfy the condition was studied by Frasin and Jahangiri [1].
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 , we have the class (ii)for , we have the class (iii)for , the class introduced by Frasin and Darus [2].
In this paper, we introduce a new general integral operator defined by where , , and for all .
Remark 2. For , , , and , we have the integral operator introduced by Ularu and Breaz [3].
The following results will be required in our investigation.
General Schwarz Lemma (see [4]). Let the function be regular in the disk , with for fixed . If has one zero with multiplicity order bigger than for , then The equality can hold only if , where is constant.
Theorem A (see [5]). If the function is regular in the unit disk , and for all , then the function is univalent in .
Theorem B (see [6]). If satisfies the condition then is univalent in .
2. Main Results
Theorem 3. Let , where satisfies the condition , and for all . If for all , then the integral operator defined by (10) is in the univalent function class .
Proof. From (10), we obtain
for . This equality implies that
By differentiating the above equality, we get
or equivalently
So we find
From the hypothesis, we have ; then by the general Schwarz lemma, we obtain that
Thus we have
Let us define the function , , . Then we obtain
for all . It follows from (26) that
Hence from (18), (25), and (27) we find
By applying Theorem A for the function , we prove that is in the univalent function class .
Setting , , , and in Theorem 3, we obtain the following consequence of Theorem 3.
Corollary 4 (see [3]). Let , where satisfies the condition, and . If then the integral operator defined by (11) is in the univalent function class .
Theorem 5. Let , where , , and for all . If for all , and if then the integral operator defined by (10) is in the class , where
Proof. From (22), we have It follows that Since , for all , applying general Schwarz lemma and using (35), we obtain This implies that the integral operator .
Setting in Theorem 5, we have the following.
Corollary 6. Let , where , , and for all . If for all , and if then the integral operator defined by (10) is in the class , where
Setting in Theorem 5, we have the following.
Corollary 7. Let , where , , and for all . If for all , and if then the integral operator defined by (10) is in the class , where
Setting , , , and in Theorem 5, we obtain the following consequence of Theorem 5.
Corollary 8 (see [3]). Let , where , , and . If then the integral operator defined by (11) is in the class , where
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.