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Journal of Mathematics
Volume 2013 (2013), Article ID 841837, 8 pages
http://dx.doi.org/10.1155/2013/841837
Research Article

Convexity and Spirallikeness Conditions for Two New General Integral Operators

1Department of Mathematics, Faculty of Science, Dicle University, 21280 Diyarbakir, Turkey
2Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 Kocaeli, Turkey

Received 3 January 2013; Accepted 11 April 2013

Academic Editor: Majid Soleimani-damaneh

Copyright © 2013 H. Özlem Güney and Serap Bulut. 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

We define two new general integral operators for certain analytic functions in the unit disc and give some sufficient conditions for these integral operators on some subclasses of analytic functions.

1. Introduction

Let denote the class of all functions of the form which is analytic in the open unit disc .

In particular, we set

The fractional derivative of order is defined, for a function , by where the function is analytic in a simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .

It readily follows from (3) that

Using the operator ,

Bulut [1] defined the general differential operator , namely, the generalization of the generalized Al-Oboudi differential operator as follows:

If is given by (1), then by (5) and (6), we see that where

By using the operator defined by (7), we introduce the new classes and as follows.

Definition 1. A function is in the class if and only if satisfies where , , , and is defined by (7).

Definition 2. A function is in the class if and only if satisfies where , , , , and is defined by (7).

Note that

Remark 3. (i) For , we have the classes introduced and studied by Bulut [1]. In particular, we get the classes introduced and studied by Bulut [2].(ii) For and , we have the classes In particular, we set for the -valently starlike and -valently convex functions with complex order , respectively.(iii) For , we have the classes In particular, we set Also, we have the classes introduced and studied by Shams et al. [3].(iv) For and , we have the classes of -valently starlike and -valently convex functions with complex order and type , respectively. In particular, we set Note that for , we have which are the classes of starlike and convex functions type , respectively.(v) For , , , and , we have the classes of uniformly starlike and uniformly convex functions of order , respectively, investigated by Rønning [4]. In particular, we get the classes of uniformly starlike and uniformly convex functions, respectively, first defined by Goodman [5].

Remark 4. If we set and in Definitions 1 and 2, then we have the classes respectively. Also, we put In particular, we set for the the class of -spirallike functions of order [6], and the class of -Robertson functions of order [7], respectively.
We now define the following general integral operators.

Definition 5. Let , , and . One defines the following general integral operators: where , , and is defined by (7) for .

Remark 6. (i) For , , , and , we have the integral operators (ii) For , , and , we have the integral operators introduced by Saltık et al. [8].(iii) For , we have the integral operators introduced by Frasin [9].

2. Sufficient Conditions for the Operator

Theorem 7. Let , , and . Also let , , , and for . If then the integral operator , defined by (29), is in the class , where

Proof. From the definition (29), we observe that . On the other hand, it is easy to see that Differentiating (38) logarithmically and multiplying by , we obtain or equivalently Then, by multiplying (40) with , we have or equivalently Since , we get Since the integral operator with

Corollary 8. In Theorem 7, letting(i), , and , one obtains [10, Theorem 1].(ii), , and , , one obtains [11, Theorem 1].(iii), , and , , one obtains [12, Theorem 1].

Putting , , , , , and in Theorem 7, we have the following.

Corollary 9. Let and . Also let , , , and . If then the integral operator , defined by (32), is in the class , where

Putting and in Corollary 9, we have the following.

Corollary 10. Let , , , , and . If then the integral operator is convex of complex order and type ; that is, .

Putting , , and in Theorem 7, we have the following.

Corollary 11. Let , , , , and for . If then the integral operator is in the class , where

Theorem 12. Let , , and . Also let , , and for . If then the integral operator , defined by (29), is in the class , where

Proof. From (24), (25), and (40), we get the desired result.

Putting , , , and in Theorem 12, we have the following.

Corollary 13. Let and . Also let , , and . If then the integral operator , defined by (32), is in the class , where

Putting and in Corollary 13, we have the following.

Corollary 14. Let , , , and . If then the integral operator , defined by (49), is in the class , where

Putting and in Theorem 12, we have the following.

Corollary 15. Let , , , and for . If then the integral operator , defined by (51), is in the class , where

Theorem 16. Let , , and . Also let , , and for . If for all , then the integral operator , defined by (29), is -valently convex of complex order ; that is, .

Proof. From (43) and (61), we easily get that the integral operator is -valently convex of complex order .

3. Sufficient Conditions for the Operator

Theorem 17. Let , , and . Also let , , , and for . If then the integral operator , defined by (31), is in the class , where

Proof. From the definition (31), we observe that . On the other hand, it is easy to see that Differentiating (64) logarithmically and multiplying by , we obtain or equivalently Then, by multiplying (66) with , we have or equivalently Since , we get Since the integral operator with

Corollary 18. In Theorem 17, letting(i), , and , one obtains [10, Theorem 3].(ii), , and , , one obtains [11, Theorem 3].(iii), , and , , one obtains [12, Theorem 2].

Putting , , , , , and in Theorem 17, we have the following.

Corollary 19. Let and . Also let , , and . If then the integral operator , defined by (33), is in the class , where

Putting and in Corollary 19, we have the following.

Corollary 20. Let , , , , and . If then the integral operator is convex of complex order and type ; that is, .

Putting , , and in Theorem 17, we have the following.

Corollary 21. Let , , , , and for . If then the integral operator is in the class , where

Theorem 22. Let , , and . Also let , , and for . If then the integral operator , defined by (31), is in the class , where

Proof. From (25) and (66), we get the desired result.

Putting , , , , and in Theorem 22, we have the following.

Corollary 23. Let and . Also let , , and . If then the integral operator , defined by (33), is in the class , where

Putting and in Corollary 23, we have the following.

Corollary 24. Let , , , and . If then the integral operator , defined by (75), is in the class , where

Putting and in Theorem 22, we have the following.

Corollary 25. Let , , , and for . If then the integral operator , defined by (77), is in the class , where

Theorem 26. Let , , and . Also let , , and for . If for all , then the integral operator , defined by (31), is -valently convex of complex order ; that is, .

Proof. From (69) and (87), we easily get that the integral operator is -valently convex of complex order .

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