Abstract

Let be the integral operator defined by , where each of the functions and is, respectively, analytic functions and functions with positive real part defined in the open unit disk for all . The object of this paper is to obtain several univalence conditions for this integral operator. Our main results contain some interesting corollaries as special cases.

1. Introduction and Definitions

Let denote the class of functions of the form which are analytic in the open unit disk is univalent in . Also, let be the class of all functions which are analytic in and satisfy , . Frasin and Darus [1] defined the family , , so that it consists of functions satisfying the condition In this paper, we obtain new sufficient conditions for the univalence of the general integral operator defined by where , , , and for all .

Here and throughout in the sequel, every multivalued functions is taken with the principal branch.

Remark 1. Note that the integral operator generalizes the following operators introduced and studied by several authors as follows.(i) For , where , we obtain the integral operator  introduced and studied by Frasin [2].(ii) For , , we obtain the integral operator  introduced and studied by Frasin [3].(iii) For , , we obtain the integral operator  introduced and studied by Frasin [4].(iv) For , , and , we obtain the integral operator  introduced and studied by Frasin [5].(v) For and , we obtain the integral operator  introduced and studied by D. Breaz and N. Breaz [6].(vi) For , , and , we obtain the integral operator  introduced and studied by D. Breaz and N. Breaz [6].(vii) For , , and , we obtain the integral operator  introduced and studied by Breaz et al. [7].(viii) For , , , and , we obtain the integral operator  studied in [8].(ix) For , , , and , we obtain the integral operator  studied in [9]. In particular, for , we obtain Alexander integral operator which was introduced in [10] as follows (x) For , , , and , we obtain the integral operator studied in [11].In order to derive our main results, we have to recall here the following lemmas.

Lemma 2 (see [12]). Let with . If satisfies for all , then the integral operator is in the class .

Lemma 3 (see [13]). Let with , with , . If satisfies for all , then the integral operator defined by (16) is in the class .

Lemma 4 (see [14]). If , then

Lemma 5 (see [9]). If , then When , so .

Lemma 6 (see [9]). If , then Also we need the following general Schwarz lemma.

Lemma 7 (see [15]). Let the function be regular in the disk , with for fixed . If has one zero with multiplicity order bigger than for , then The equality holds only if where is constant.

Lemma 8 (see [16]). If , then

2. Univalence Conditions for the Operator

We first prove the following theorem.

Theorem 9. Let , , for all and with . If then the integral operator defined by (3) is in the class .

Proof. Define the regular function by Then it is easy to see that and . Differentiating both sides of (26) logarithmically, we obtain Thus, we have Since , for all , from (28), (18), (19), and (20), we obtain Multiplying both sides of (29) by , we get for all .
Let us denote , , , and . It is easy to prove that From (31), (30), and the hypotheses (24), we have for all . Applying Lemma 2 for the function , we prove that .

Letting , , , , and in Theorem 9, we obtain the following corollary.

Corollary 10. Let , , , and all with . If and then the integral operator defined by is in the class .

If we set in Corollary 10, we have the following.

Corollary 11. Let , and all with . If then the integral operator defined by (34) is in the class .

Next, we prove the following theorem.

Theorem 12. Let for all and each satisfies condition (15) with and and then, for any complex number with , the integral operator defined by (4) is in the class .

Proof. Suppose that for all . Thus we have where for all . Differentiating both sides of (37) logarithmically, we obtain Define the regular function as in (25). Thus from (27) we have and so From Lemma 8, it follows that Multiplying both sides of (41) by , from Lemma 5 with , we get Suppose that . Define a function by Then is an increasing function and consequently, for ; , we obtain We thus find from (42) and (44) that Using the hypotheses (36) for , we readily get Now if , we define a function by We observe that the function is decreasing and consequently, for , we have for all . It follows from (40) and (42) that Using once again the hypotheses (36) when , we easily get Finally by applying Lemma 2, we conclude that the integral operator defined by (4) is in the class .

Letting , , , , and in Theorem 12, we obtain the following corollary.

Corollary 13. Let , and satisfies condition (15). If then, for any complex number with , the integral operator defined by (4) is in the class .

Using Lemma 3, we derive the following theorem.

Theorem 14. Let for all , , and each satisfies condition (15). If then, for any complex number with , the integral operator defined by (4) is in the class .

Proof. From (40), we have Suppose that . Define a function by Then is an increasing function and consequently for ; , we obtain We thus find from (53) that Using the hypotheses (52) for , we readily get Now if , we define a function by We observe that the function is decreasing and consequently for ; , and using once again the hypotheses (36) when , we easily get Finally, by applying Lemma 3, we conclude that .

Conflict of Interests

The authors declare that they have no conflict interests.

Authors’ Contribution

The first author is currently a Ph.D. student under supervision of the second author and jointly worked on deriving the results. All authors read and approved the paper.

Acknowledgment

The work presented here was partially supported by ERGS/1/2013/STG06/UKM/01/2 and DIP-2013-1.