Abstract and Applied Analysis

Volume 2013 (2013), Article ID 621810, 6 pages

http://dx.doi.org/10.1155/2013/621810

Research Article

On General Integral Operator of Analytic Functions

Nasser Alkasbi and Maslina Darus

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 3 August 2013; Accepted 27 October 2013

Academic Editor: Mohamed Kamal Aouf

Copyright © 2013 Nasser Alkasbi and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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