Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function
We define a new general integral operator for meromorphic functions involving the generalized hypergeometric function. Furthermore, we study the characterization and other properties for this operator.
Let denote the class of functions of the form which are analytic in the punctured open unit disk .
We say that a function is meromorphic starlike of order and belongs to the class , if it satisfies the following inequality: A function is a meromorphic convex function of order , if satisfies the following inequality: and we denote this class by .
A -hypergeometric function is a power series in one complex variable with power series coefficients which depend (apart from ) on complex upper parameters , , as follows: with , where when , , and is the Pochhammer symbol defined by Tracing back the history of basic hypergeometric series defined by (5) has brought us to Heine dated in 1846. Therefore it is sometimes called Heine’s series. For brief survey on -hypergeometric functions, one may refer to [3–5] (see also [6–8]).
For , , and , the basic hypergeometric function defined in (5) takes the form which converges absolutely in the open unit disk .
Corresponding to the function , let Analogous to the differential operator defined in  which involves the -hypergeometric functions on the normalized analytic functions, we define the following differential operator on the space of meromorphic functions in the class by If , then from (9) we may deduce that where , and .
It should be remarked that the operator given by (10) is a generalization of many other operators considered earlier; for example, consider the following.(i)For , , , , the operator was defined by Frasin and Darus  and studied by El-Ashwah and Aouf .(ii)For , , , , , , the operator was imposed by Liu and Srivastava .(iii)For , , , , , the operator was introduced and studied by Liu and Srivastava . Further, we note in passing that this operator is closely related to the Carlson-Shaffer operator defined on the space of analytic univalent functions in .(iv)For , , , , , the operator , where is the differential operator, was introduced by Ganigi and Uralegaddi , and then it was generalized by Yang .Now, making use of the differential operator given by (10), we introduce the following integral operator in the class .
Definition 1. Let , . We define the integral operator by
With the aid of differential operator given by (10), we define a new subclass of functions in as follows.
Definition 3. Let a function be analytic in . Then is in the class if, and only if, satisfies where is defined in (10) and .
Definition 4. Let a function be analytic in . Then is in the class if, and only if, satisfies where is defined in (10) and , .
Definition 5. Let a function be analytic in . Then is in the class if, and only if, satisfies where is defined in (10) and .
Definition 6. Let a function be analytic in . Then is in the class if, and only if, satisfies where is defined in (10) and .
2. Preliminary Definitions
We begin by recalling each of the following definitions of subclasses of meromorphic functions , and which will be required in our investigation.
Definition 8 (see ). Let a function be analytic in . Then is in the class if, and only if, satisfies where , .
Definition 9 (see ). Let a function be analytic in . Then is in the class if, and only if, satisfies where .
Definition 10 (see ). Let a function be analytic in . Then is in the class if, and only if, satisfies where .
3. Main Results
Theorem 11. Let , for and . If and , then is in the class , .
Proof. By differentiating the equality given by (12), we get Again, differentiating (21), we deduce that By using (21) and (23), we have Multiplying (24) by yields that or, equivalently, that Thus, we get That is, Taking the real parts of both terms of the last expression and using the fact that for all , we impose Setting and since we conclude that Hence, by (20), we have , so that , which evidently completes the proof of Theorem 11.
Theorem 12. Let , for , and . If then is in the class , where .
Proof. We know from the proof of Theorem 11 that which is equivalent to Taking the real part of both terms of (35), we get Now, since , we have Letting and in view of (33), it follows that , so that . This completes the proof of Theorem 12.
Theorem 13. Let , for , and . If then is in the class .
Theorem 14. Let for , and . If then is in the class .
Proof. Our aim is to prove that With the aid of (26), (36) can be written as follows: Since , it follows from Definition 6 and by a simple calculation that which means that , and we obtain the proof of the theorem.
Conflict of Interests
The authors declare that they have no competing interests.
Ibtisam Aldawish and Maslina Darus read and approved the final paper.
The work presented here was partially supported by UKM’s Grant FRGSTOPDOWN/2013/ST06/UKM/01/1.
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