Abstract
We introduce new integral operators of analytic functions and defined in the open unit disk . For these operators, we discuss some univalence conditions.
1. Introduction and Preliminaries
Let denote the class of all functions of the form which are analytic in the open unit disk and satisfy the following usual normalization condition: Also, let denote the subclass of consisting of functions , which are univalent in (see, for details [1]; see also [2, 3]).
In [4, 5], Pescar gave the following univalence conditions for the functions .
Theorem 1.1 (see [4]). Let be a complex number, , and a complex number, , and a regular function in . If for all , then the function is regular and univalent in .
Theorem 1.2 (see [5]). Let be a complex number, , and a complex number, , and . If for all , then for any complex number , the function is in the class .
On the other hand, for the functions , Ozaki and Nunokawa [6] proved another univalence condition asserted by Theorem 1.3.
Theorem 1.3 (see [6]). Let satisfy the condition Then is univalent in .
In the paper [7], Pescar determined some univalence conditions for the following integral operators.
Theorem 1.4 (see [7]). Let the function satisfy (1.8), a positive real number fixed, and a complex number. If , for all , then the function is in the class .
Theorem 1.5 (see [7]). Let a real number, , and a complex number, . If then the function is in the class .
Theorem 1.6 (see [7]). Let satisfies (1.8), a complex number, fixed, , and a complex number, . If for all , then for any complex number the function is in the class .
In this paper, we introduce the following integral operators as follows:
Remark 1.7. For and , the integral operators (1.15), (1.16), and (1.17) would reduce to the integral operators (1.10), (1.12), and (1.14).
In this paper, we generalize the integral operators given by Pescar [7], and we study the univalence conditions for the integral operators defined by (1.15), (1.16), and (1.17).
For this purpose, we need the following result.
Lemma 1.8 (General Schwarz Lemma [8]). 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.
2. Main Results
Theorem 2.1. Let , where satisfies the condition (1.8), and are real positive numbers, and a complex number, . If then the integral operator defined by (1.15) is in the class .
Proof. From (1.15), we have Let us consider the function The function is regular in . From (2.4), we get Also, a simple computation yields which readily shows that From the hypothesis of Theorem 2.1, we have then by General Schwarz Lemma for the function , we obtain Using the inequality (2.7), we have From (2.10) and since satisfies the condition (1.8), we have from which, by (2.2), we get Applying Theorem 1.1, we conclude that the integral operator defined by (1.15) is in the class .
Setting and in Theorem 2.1, we immediately arrive at the following application of Theorem 2.1.
Corollary 2.2. Let , where satisfies the condition (1.8) and a complex number, . If then the integral operator defined by (1.15) is in the class .
Theorem 2.3. Let , where satisfies the inequality . Also, let be a real number, , and a complex number with If then the integral operator defined by (1.16) is in the class .
Proof. We observe that Let us consider the function The function is regular in . From (2.17), we have which readily shows that From (2.19) and the conditions of Theorem 2.3, we get Applying Theorem 1.1, we conclude that the integral operator defined by (1.16) is in the class .
Setting in Theorem 2.3, we obtain the following consequence of Theorem 2.3.
Corollary 2.4. Let , where satisfies the condition a real number, , and a complex number with . If then the integral operator defined by (1.16) is in the class .
Theorem 2.5. Let , where satisfies the condition (1.8), a complex number, and are real positive numbers, and a complex number, . If then for any complex number , the integral operator defined by (1.17) is in the class .
Proof. Let us consider the function The function is regular in . From (2.24), we have which readily shows that By the General Schwarz Lemma for the function , we obtain and using the inequality (2.26), we have From (2.28) and since satisfies the condition (1.8), we get From (2.23), we have and using (2.29), we obtain Applying Theorem 1.2, we conclude that the integral operator defined by (1.17) is in the class .
Setting and in Theorem 2.5, we obtain the following corollary.
Corollary 2.6. Let , where satisfies the condition (1.8), a complex number, , and a complex number, . If then for any complex number , the integral operator defined by (1.17) is in the class .
Acknowledgments
This paper was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007–2013 cofinanced by the European Social Fund-Investing in People.