International Journal of AnalysisVolume 2013 (2013), Article ID 380938, 4 pageshttp://dx.doi.org/10.1155/2013/380938

Research Article

Some Properties for Certain Class of Analytic Functions with Varying Arguments

M. K. Aouf, A. O. Mostafa, A. Shamandy, and E. A. Adwan

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 21 December 2012; Accepted 11 March 2013

Academic Editor: Tohru Ozawa

Copyright © 2013 M. K. Aouf et al. 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 introduce a new class of analytic functions with varying arguments in the open unit disc defined by the Salagean operator. The object of the present paper is to determine coefficient estimates, extreme points, and distortion theorems for functions belonging to the class .

1. Introduction

Let denote the class of functions of the form which are analytic in the open unit disc . If and are analytic functions in , we say that is subordinate to , written if there exists a Schwarz function , which (by definition) is analytic in with and for all , such that . Furthermore, if the function is univalent in , then we have the following equivalence (cf, e.g., [1, 2]): For function , Salagean [3] defined the differential operator as follows: For , and for all , let denotes the subclass of consisting of functions of the form (1) and satisfying the analytic criterion: It is noticed that for suitable choices of , and , we obtain the following subclasses studied by various authors:(1) (see Sivasubramanian et al. [4], with );(2) (see Aouf et al. [5], with );(3) (see Chunyi and Owa [6]);(4) (see Chen [7, 8] and Goel [9]);(5) (see Srivastava and Owa [10]).

Also we note the following.(1)Putting in (4), the class reduces to the class ,.

Definition 1 (see [11]). A function defined by (1) is said to be in the class if and for all . If, furthermore, there exists a real number such that then is said to be in the class . The union of taken over all possible sequences and all possible real numbers is denoted by .Let denote the subclass of consisting of functions in .We note the following:(1) (see Aouf et al. [12]);(2) (see Srivastava and Owa [10]);(3) (see Srivastava and Owa [10]).

Also we note that(i) denotes the subclass of , consisting of functions belonging to the class ;(ii) denotes the subclass of , consisting of functions belonging to the class .

2. Coefficient Estimates

Unless otherwise mentioned, we will assume in the reminder of this paper that and .

Theorem 2. Let the function be of the form (1), if then .

Proof. A function of the form (1) belongs to the class if and only if there exists a function , such that or, equivalently, Thus, it is sufficient to prove that Indeed, letting we have In view of (6), the last inequality is less than zero; hence . This completes the proof of Theorem 2.

Theorem 3. Let the function be of the form (1), then is in the class if and only if

Proof. In view of Theorem 2, we need only to show that each function from the class satisfies the coefficient inequality (6). Let . Then, by (8) and (1), we have Since , lies in the class for some sequence and a real number such that , set in the above inequality and since , then It is clear that the denominator of the left-hand said cannot vanish for . Moreover, it is positive for and in consequence for . Thus, we have which, upon letting, readily yields the assertion (11). This completes the proof of Theorem 3.

Corollary 4. Let the function defined by (1) be in the class, then The result is sharp for the function

3. Distortion Theorems

Theorem 5. Let the function defined by (1) be in the classThen The result is sharp.

Proof. Since is an increasing function of , from Theorem 2, we have that is, Thus, Similarly, we get Finally, the result is sharp for the function at .This completes the proof of Theorem 5.

Corollary 6. Under the hypotheses of Theorem 5, is included in a disc with center at the origin and radius given by

Theorem 7. Let the function defined by (1) belong to the classThen The result is sharp for the function given by (23) at .

Proof. Since , where given by (18) is increasing function of , in view of Theorem 3, we have that is, Thus, Similarly, we get Finally, the result is sharp for the function given by (23). This completes the proof of Theorem 7.

Corollary 8. Let the function defined by (1) be in the class . Then is included in a disc with center at the origin and radius given by

4. Extreme Points

Theorem 9. Let the function defined by (1) be in the class , with where . Define and Then is in the class if and only if it can be expressed in the form where and .

Proof. If with and , then Hence, . Conversely, let the function defined by (1) belong to the class , and define From Theorem 3, and so . Since , then This completes the proof of Theorem 9.

Remark 10. Specializing , and in the above results, we obtain the corresponding results for the corresponding classes and defined in Section 1.

Acknowledgments

The authors would like to thank the referees of the paper for their helpful suggestions.

References

T. Bulboaca, Differential Subordinations and Superordinations, Recent Results, House of Scientific Book, Cluj-Napoca, Romania, 2005.

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 Inc., New York, NY, USA, 2000.

G. Sãlãgean, “Subclasses of univalent functions,” in Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983.

S. Sivasubramanian, A. Mohammed, and M. Darus, “Certain subordination properties for subclasses of analytic functions involving complex order,” Abstract and Applied Analysis, vol. 2011, Article ID 375897, 8 pages, 2011.

M. K. Aouf, A. Shamandy, A. O. Mostafa, and E. A. Adwan, “Subordination results for certain classes of analytic functions defined by convolution with complex order,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 1, pp. 61–68, 2011.

G. Chunyi and S. Owa, “Certain class of analytic functions in the unit disk,” Kyungpook Mathematical Journal, vol. 33, no. 1, pp. 13–23, 1993.

M. P. Chen, “On the regular functions satisfying ,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 3, no. 1, pp. 65–70, 1975.

M. P. Chen, “On functions satisfying ,” Tamkang Journal of Mathematics, vol. 5, pp. 231–234, 1974.

R. M. Goel, “On functions satisfying ,” Publicationes Mathematicae Debrecen, vol. 18, pp. 111–117, 1971.

H. M. Srivastava and S. Owa, “Certain classes of analytic functions with varying arguments,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 217–228, 1988.

H. Silverman, “Univalent functions with varying arguments,” Houston Journal of Mathematics, vol. 7, no. 2, pp. 283–287, 1981.

M. K. Aouf, A. Shamandy, and A. A. Attiyia, “On certain classes of analytic functions with varying arguments,” Chinese Journal of Mathematics, vol. 24, no. 2, pp. 135–143, 1996.