SUBORDINATION RESULTS OF MULTIVALENT FUNCTIONS DEFINED BY CONVOLUTION

Abstract

SUBORDINATION RESULTS OF MULTIVALENT FUNCTIONS DEFINED BY CONVOLUTION

Abstract

Using the method of differential subordination, we investigate some properties of certain classes of multivalent functions, which are defined Analytic functionsby means of convolution.

Keywords

• Analytic functions,Multivalent functions,Convolution product,Differential subordination,2000 AMS Classification: 30 C 45

References

1. Al-Oboudi, F. M. On univalent functions defined by a generalized S˘al˘agean operator, Int. J. Math. Math. Sci. 27, 1429–1436, 2004.
2. Al-Oboudi, F. M. and Al-Amoudi, K. A. On classes of analytic functions related to conic domains, J. Math. Anal. Appl. 339 (1), 655-ˆu667, 2008.
3. Aouf, M. K. and Mostafa, A. O. On a subclass of n–p–valent prestarlike functions, Comput. Math. Appl. 55 (4), 851–861, 2008.
4. Aouf, M. K. and Mostafa, A. O. Sandwich theorems for analytic functions defined by con- volution, Acta Univ. Apulensis 21, 7–20, 2010.
5. Bernardi, S. D. Convex and univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1996.
6. Carlson, B. C. and Shaffer, D. B. Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15, 737–745, 1984.
7. C˘ata¸s, A. On certain classes of p–valent functions defined by multiplier transformations, in Proc. of the International Symposium on Geometric Function Theory and Applications, Istanbul, Turkey, 241–250, 2007.
8. Cho, N. E. and Kim, T. G. Multiplier transformations and strongly close-to-convex func- tions, Bull. Korean Math. Soc. 40 (3), 399–410, 2003.

9. Dziok, J. and Srivastava, H. M. Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103, 1–13, 1999.
10. Dziok, J. and Srivastava, H. M. Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math. 5, 115–125, 2002.
11. Dziok, J. and Srivastava, H. M. Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transf. Spec. Funct. 14, 7–18, 2003.
12. Hallenbeck, D. J. and Ruscheweyh, St. Subordination by convex functions, Proc. Amer. Math. Soc. 52, 191–195, 1975. [13] Hohlov, Yu. E. Operators and operations in the univalent functions (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. 10, 83–89, 1978.
13. Kamali, M. and Orhan, H. On a subclass of certain starlike functions with negative coeffi- cients, Bull. Korean Math. Soc. 41 (1), 53–71, 2004.
14. Libera, R. J. Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16, 755– 658, 1965.
15. Livingston, A. E. On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17, 352–357, 1966. [17] MacGregor, T. H. Radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14, 514–520, 1963.
16. Miller, S. S. and Mocanu, P. T. Differential subordinations and univalent functions, Michigan Math. J. 28 (2), 157–171, 1981. [19] Miller, S. S. and Mocanu, P. T. Differential Subordination: Theory and Applications (Series on Monographs and Textbooks in Pure and Applied Mathematics 225, Marcel Dekker Inc., New York and Basel, 2000).
17. Owa, S. and Srivastava, H. M. Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39, 1057–1077, 1987.
18. Patel, J. Radii of γ–spirallikeness of certain analytic functions, J. Math. Phys. Sci. 27, 321–334, 1993.
19. Ruscheweyh, St. New criteria for univalent functions, Proc. Amer. Math. Soc. 49, 109–115, 1975.
20. Saitoh, H. A linear operator and its applications of first order differential subordinations, Math. Japon. 44, 31–38, 1996. [24] S˘al˘agean, G. S. Subclasses of Univalent Functions, Lecture Notes in Math. (Springer-Verlag, Berlin) 1013, 362–372, 1983. [25] Singh, R. and Singh, S. Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc. 106, 145–152, 1989.
21. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth Edition(Cambridge University Press, Cam- bridge, 1927).