Abstract
We study univalence properties for certain subclasses of univalent functions , , , and , respectively. These subclasses are associated with a generalized integral operator. The extended Becker-typed univalence criteria will be studied for these subclasses.
1. Introduction and Preliminaries
Let denote the class of analytic functions in the open unit disk normalized by . Thus, each has a Taylor series representation Let be the subclass of consisting of functions of the form Let be the univalent subclass of which satisfies Let be the subclass of for which . Let be the subclass of consisting of functions of the form (1.2) which satisfy Next, we define a subclass of consisting of all functions that satisfy For functions and , the Hadamard product (or convolution) is defined as usual by Define the function by where is the famous Pochhammer symbol defined in terms of Gamma function. It is easily seen that is a convex function, since
Using the fractional derivative of order , [1], Owa and Srivastava [2] introduced the operator which is known as an extension of fractional derivative and fractional integral, as follows: Note that
For a function in , we define , the linear fractional differential operator, as follows: If is given by (1.1), then by (1.8) and (1.9), we see that From (1.8) and (1.9), can be written in terms of convolution as where which generalizes many operators. Indeed, if we choose suitably values of , , , and in (1.12), we have the following.(i), , and , we obtain given by Aouf et al. [3].(ii), , , and , we obtain given by Al-Oboudi [4]. (iii), , , , and , we obtain given by Sălăgean [5].(iv), , , , and , we obtain given by Uralegaddi and Somanatha [6].(v), , , and , we obtain given by Cho and Srivastava [7] and Cho and Kim [8].(vi), , , , and , we obtain Owa and Srivastava differential operator [2].(vii), , and , we obtain given by Al-Oboudi and Al-Amoudi [9, 10].(viii), , and , we obtain given by Catas [11].(ix), , , and , we obtain given by Kumar et al. and Srivastava et al., respectively [12, 13].
Next, we introduce a new family of integral operator by using generalized differential operator already defined above.
For and , , we define a family of integral operators by which generalize many integral operators. In fact, if we choose suitable values of parameters in this type of operator, we get the following interesting operators.(i), , , , , and , we obtain given by Bulut [14].(ii), , , , , , and , we obtain given by Breaz et al. [15].(iii), , , , , , and , we obtain given by D. Breaz and N. Breaz [16].
For our main result, we need the following lemmas.
Lemma 1.1 (see [17, 18]). Let be a complex number, , . If is a regular function in and then the function is regular and univalent in .
Lemma 1.2 (Schwarz Lemma). Let the function be regular in the disk with . If has one zero with multiply for , then and equality holds only if , where is constant.
Lemma 1.3 (see [19]). Let be a complex number with such that . If satisfies the condition then the function is analytic and univalent in .
Lemma 1.4 (see [20]). If a function , then
2. Univalence Properties
In this section, we will discuss the univalence properties of the new family of integral operators mentioned above.
Theorem 2.1. Let be a complex number, , for all and for such that where , are complex numbers. If then the family is univalent.
Proof. Since , so by Lemma 1.4, we have Now, by using hypothesis, we have so by Lemma 1.3, we get Let so Let which implies that This implies that or Using (2.5), we get This implies that By using (2.3), we get which implies that because implies that Now, we calculate This implies that By using (2.46), we conclude that Hence, by Lemma 1.3, the family of integral operators is univalent.
Corollary 2.2. Let be a complex number, , for all and , , for all such that where , are complex numbers. If then the family is univalent.
Corollary 2.3. Let be a complex number, , , for all and the family , , , for all such that where , are complex numbers. If then the family is univalent.
Using the method given in the proof of Theorem 2.1, one can prove the following results.
Theorem 2.4. Let be a complex number, , for all and the family for and such that where , are complex numbers. If then the family is univalent.
Theorem 2.5. Let be a complex number, , for all and for such that where , are complex numbers. If then the family is univalent.
Theorem 2.6. Let be a complex number, , for all and for such that where , are complex numbers. If then the family is univalent.
Theorem 2.7. Let be a complex number, , for all and , for such that where , are complex numbers. If then the family is univalent.
Proof. Using the proof of Theorem 2.1, we have Since , so by using (1.4), we get So from (2.33), we get or Now, we evaluate the expression Using (2.45) and (2.46), we conclude that Hence by using Lemma 1.3, the family is univalent.
Corollary 2.8. Let be a complex number, , for all and , for such that where , are complex numbers. If then the family is univalent.
Corollary 2.9. Let be a complex number, , , for all and , for such that where , are complex numbers. If then the family is univalent.
Using a similar method as in the proof of Theorem 2.7, one can prove the following results.
Theorem 2.10. Let be a complex number, , for all and , for such that where , are complex numbers. If then the family is univalent.
Theorem 2.11. Let be a complex number, , for all and , for such that where , are complex numbers. If then the family is univalent.
Note that some other related work involving integral operators regarding univalence criteria can also be found in [21–23].
Acknowledgment
The work presented here was partially supported by UKM-ST-06-FRGS0244-2010.