SOME CONVEX FAMILIES OF MULTIVALENTLY ANALYTIC FUNCTIONS DEFINED BY USING CERTAIN OPERATORS OF FRACTIONAL DERIVATIVES

Year 2011, Volume: 4 Issue: 2, 135 - 144, 12.03.2014

Nihat Yağmur , Ömer Çetin Hüseyin Irmak

Abstract

Making use of fractional derivative operator, we introduce a general class  of functions which are analytic and p-valent in the open unit disk U, and obtain a necessary and sufficient condition for a function to be in the class , distortion bounds, inclusion property and the radii of p-valently close-to-convexity, p-valently starlikeness, p-valently convexity for this generalized class of p-valent function.

2000 Mathematics Subject Classification. Primary 30C45, 26A33, Secondary 33C05

Keywords

P-değerli, kesirsel integral, kesirsel türev, büyüme

References

• Altıntaş, O., Irmak H. and Srivastava, H. M. 1995. A Subclass of Analytic Functions Defined by Using Certain Operators of Fractional Calculus, Comput. Math. Appl. 30, 1-9.
• Altıntaş, O., Irmak H. and Srivastava, H. M. 1995. fractional Calculus and Certain Starlike Functions With Negative Coefficients, Comput Math. Appl. 30 (2), 9-15.
• Bhoosnurmath, S. S. and Swamy, S. R. 1985. Certain Classes of Analytic Functions With Negative Coefficients, Indian J. Math. 27, 89-98.
• Chen, M. -P., Irmak H. and Srivastava, H. M. 1996. Some Multivalently Functions With Negative Coefficients Defined by Using a Differantial Operator, Pan Amer. Math. J.6 (2), 55-64.
• Chen, M. -P., Irmak H. and Srivastava, H. M. 1997. Some Families of Multivalently Analytic Functions With Negative Coefficients, J. Math. Anal. Appl. 214 , 674-690.
• Chen, M. -P., Irmak, H. and Srivastava, H. M. 1998. A Certain Subclass of Analytic Functions Involving Operators of Fractional Calculus, Comput. Math. Appl. 35 (5) , 83-91.
• Duren, P. L. 1983. Univalent Functions, Grunlehren der Mathematischen Wissenschaften, Bd. 259, Springer-Verlag, New York, Berlin, Heiderberg, and Tokyo.
• Goodman, A. W. 1983. Univalent Functions, Vols. I and II, Polygonal Publishing Conpany Washington, New Jersey.
• Irmak, H. 1995. Certain Subclasses of P-Valently Starlike Functions With Negative Coefficients, Bull. Calcutta Math. Soc. 87, 589-598.
• Irmak, H., Lee, S. H. and Cho, N. E. 1997. Some Multivalently Analytic Functions With Negative Coefficients and Their Subclasses Defined by Using a Differantial Operators, Kyungpook Math. J. 37 (1), 43-51.
• Owa, S. 1978. On the Distortion Theorems. I, Kyungpook Maht. J., 18,53-59.
• Owa, S. and Srivastava, H. M. 1987. Univalent and Starlike Generalized Hypergeometric Function, Canad. J. Math., 39, 1057-1077.
• Srivastava, H. M. and Owa, S. 1992. (Editors), Current Topics in Analitic Function Theory, Word Scientific Publishing Company, Singapure, New Jersey, London, and HongKong.