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`Journal of MathematicsVolume 2013 (2013), Article ID 784560, 6 pageshttp://dx.doi.org/10.1155/2013/784560`
Research Article

## Certain Properties of Multivalent Functions Associated with the Dziok-Srivastava Operator

Department of Mathematics, Faculty of Science, Fayoum 63514, Egypt

Received 11 October 2012; Accepted 26 October 2012

Copyright © 2013 T. M. Seoudy. 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

By making use of the techniques of the differential subordination, we derive certain properties of -valent functions associated with the Dziok-Srivastava operator.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disk  . We write .

Suppose that and are analytic in . We say that the function is subordinate to in , or superordinate to in , and we write or , if there exists an analytic function in with and , such that . If is univalent in , then the following equivalence relationship holds true (see [13]):

For functions given by we define the Hadamard product (or convolution) of and by

For complex parameters and , the generalized hypergeometric function is defined (see [4]) by the following infinite series: where is the Pochhammer symbol defined, in terms of the Gamma function , by Corresponding a function defined by Dziok and Srivastava [5] considered a linear operator defined by the following Hadamard product: If is given by (1), then we have where To make the notation simple, we write It easily follows from (9) or (10) that It should be remarked that the linear operator is a generalization of many other linear operators considered earlier. In particular, for we have the following observations:(i), where the linear operator was investigated by Hohlov [6];(ii), where the linear operator was studied by Goel and Sohi [7]. In the case when , is the Ruscheweyh derivative of (see [8]);(iii), where is the generalized Bernardi-Libera-Livingston integral operator (see [9]);(iv), where is the fractional integral of of order when and fractional derivative of of order when . The extended fractional differintegral operator was introduced and studied by Patel and Mishra [10]. The fractional differential operator with was investigated by Srivastava and Aouf [11]. The operator was introduced by Owa and Srivastava [12] (see also [1315]).(v) = , where the linear operator was studied by Saitoh [16] which yields the operator introduced by Carlson and Shaffer [17] for ;(vi) = , where is the Choi-Saigo-Srivastava operator [9] which is closely related to the Carlson-Shaffer [17] operator ;(vii) = , where the operator was considered by Liu and Noor [18];(viii) = , where is the Cho-Kwon-Srivastava operator [19].

In recent years, many interesting subclasses of analytic functions, associated with the Dziok-Srivastava operator and its many special cases, were investigated by, for example, Dziok and Srivastava [5, 20], Gangadharan et al. [21], Liu and Noor [18], Liu [22], Liu and Srivastava [23], and others see also [19, 2426]). In the present paper, we shall use the method based upon the differential subordination to derive inclusion relationships and other interesting properties and characteristics of the Dziok-Srivastava operator .

#### 2. Main Results

Unless otherwise mentioned, we assume throughout the sequel that ; ; ; and .

Let denote the class of functions of the form that are analytic in , we write . In our present investigation, we shall require the following lemmas.

Lemma 1 (see [2]). Let be analytic and convex (univalent) in with and . If then, for and , and is the best dominant.

Lemma 2 (see [1]). Let be a set in the complex plane and be a complex number satisfying . Suppose that the function satisfies the condition for all real and for all . If the functions and , then in .

Lemma 3 (see [27]). Let be analytic in with and for all . If there exist two points such that for some and and for all , then where

Theorem 4. Let , . Let , then implies The bound is the best possible.

Proof. It easily follows from (13) that From (20) and (22), we have That is, Let then (24) may be written as By using a well-known result (see [28]) to (26) we obtain that or, equivalently, where is analytic in , and for . Since for and , (28) yields To see that the bound cannot be increased, we consider the function Since we easily have that satisfies (20) and as . This completes the proof of Theorem 4.

Theorem 5. Let , . If satisfies the following inequality then where is the positive root of the equation

Proof. Let then is analytic in and . Differentiating (36) and using (22), we obtain that where Using (33) and (38), we have Now for all real , we have where is the positive root of (35).
Note that for , , and we have and . This shows . Hence for each , . By Lemma 2, we get , and this proves (34).

Theorem 6. Suppose that ; and , . If given by satisfies then where and are the solution of the equations: where is given by (19).

Proof. Using (42) and the identity (22), it follows that for . Putting On differentiating (47) followed by a simple calculation, we get Let be the function which maps onto the angular domain with . By using (43) in (48), we get Further, an application of Lemma 1 yields in and hence for .
Suppose there exist two points such that the condition (28) is satisfied. Then by Lemma 3, we obtain (18) under the constraint (19). Therefore, we have which contradicts the assumption (43). This proves the assertion (44) of the Theorem 6.
For , Theorem 6 reduces to the following corollary.

Corollary 7. Suppose that and . If defined by (42) satisfies then where is the solution of the equation:

#### References

1. S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.
2. S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.
3. S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables, vol. 48, no. 10, pp. 815–826, 2003.
4. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, UK, 1985.
5. J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,” Applied Mathematics and Computation, vol. 103, no. 1, pp. 1–13, 1999.
6. Yu. E. Hohlov, “Operators and operations in the class of univalent functions,” Izvestiya Vysšhikh Učhebnykh Zavedenii. Matematika, vol. 10, pp. 83–89, 1978 (Russian).
7. R. M. Goel and N. S. Sohi, “A new criterion for p-valent functions,” Proceedings of the American Mathematical Society, vol. 78, no. 3, pp. 353–357, 1980.
8. S. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.
9. J. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral operators,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432–445, 2002.
10. J. Patel and A. K. Mishra, “On certain subclasses of multivalent functions associated with an extended fractional differintegral operator,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 109–122, 2007.
11. H. M. Srivastava and M. K. Aouf, “A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients I and II,” Journal of Mathematical Analysis and Applications, vol. 171, no. 1, pp. 1–13, 1992, ibid, vol. 192, pp. 673–688, 1995.
12. S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.
13. J. Dziok, “Classes of analytic functions involving some integral operator,” Folia. Scientiarum Universitatis Technicae Resoviensis, no. 20, pp. 21–39, 1996.
14. J. L. Liu and J. Patel, “Certain properties of multivalent functions associated with an extended fractional differintegral operator,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 703–713, 2008.
15. S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978.
16. H. Saitoh, “A linear operator and its applications of first order differential subordinations,” Mathematica Japonica, vol. 44, no. 1, pp. 31–38, 1996.
17. B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.
18. J.-L. Liu and K. I. Noor, “Some properties of Noor integral operator,” Journal of Natural Geometry, vol. 21, no. 1-2, pp. 81–90, 2002.
19. N. E. Cho, O. S. Kwon, and H. M. Srivastava, “Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 470–483, 2004.
20. J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,” Integral Transforms and Special Functions, vol. 14, no. 1, pp. 7–18, 2003.
21. A. Gangadharan, T. N. Shanmugam, and H. M. Srivastava, “Generalized hypergeometric functions associated with K-uniformly convex functions,” Computers and Mathematics with Applications, vol. 44, no. 12, pp. 1515–1526, 2002.
22. J.-L. Liu, “Strongly starlike functions associated with the Dziok–Srivastava operator,” Tamkang Journal of Mathematics, vol. 35, p. 37–42, 2004.
23. J. L. Liu and H. M. Srivastava, “Certain properties of the Dziok-Srivastava operator,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 485–493, 2004.
24. K. I. Noor, “Some classes of p-valent analytic functions defined by certain integral operator,” Applied Mathematics and Computation, vol. 157, no. 3, pp. 835–840, 2004.
25. K. I. Noor and M. A. Noor, “On Integral Operators,” Journal of Mathematical Analysis and Applications, vol. 238, no. 2, pp. 341–352, 1999.
26. H. M. Srivastava and J. Patel, “Some subclasses of multivalent functions involving a certain linear operator,” Journal of Mathematical Analysis and Applications, vol. 310, no. 1, pp. 209–228, 2005.
27. N. Takahashi and M. Nunokawa, “A certain connection between starlike and convex functions,” Applied Mathematics Letters, vol. 16, no. 5, pp. 653–655, 2003.
28. T. J. Suffridge, “Some remarks on convex maps of the unit disk,” Duke Mathematical Journal, vol. 37, pp. 775–777, 1970.