Surveys in Mathematics and its Applications

ISSN1842-6298 (electronic), 1843-7265 (print)

Volume 5 (2010), 35 -- 47## SOME APPLICATIONS OF GENERALIZED RUSCHEWEYH DERIVATIVES INVOLVING A GENERAL FRACTIONAL DERIVATIVE OPERATOR TO A CLASS OF ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS I

## Waggas Galib Atshan and S. R. Kulkarni

Abstract. For certain univalent function f, we study a class of functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfyingRe { (zJ A necessary and sufficient condition for a function to be in the class A_{1}^{λ, μ}f(z))^{'})/((1 - γ) J_{1}^{λ, μ}f(z) + γ z^{2}(J_{1}^{λ, μ}f(z))^{"})} > β._{γ}^{λ, μ, ν}(n, β) is obtained. In addition, our paper includes distortion theorem, radii of starlikeness, convexity and close-to-convexity, extreme points. Also, we get some results in this paper.2000 Mathematics Subject Classification:30C45.

Keywords:Distortion theorem; Radii of starlikeness; Extreme points.

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Waggas Galib Atshan S. R. Kulkarni Department of Mathematics, Department of Mathematics, College of Computer Science and Mathematics, Fergusson College, Pune - 411004, University of AL-Qadisiya, Diwaniya, Iraq. India. e-mail: waggashnd@yahoo.com e-mail: kulkarni _{-}ferg@yahoo.com