Some applications of generalized Ruscheweyh derivatives involving a general fractional derivative operator to a class of analytic functions with negative coefficients I

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, satisfying <CENTER>Re { (zJ<SUB>1</SUB><SUP>λ, μ</SUP> f(z))<SUP>'</SUP>)/((1 -γ) J<SUB>1</SUB><SUP>λ, μ</SUP> f(z) + γ z<SUP>2</SUP>(J<SUB>1</SUB><SUP>λ, μ</SUP> f(z))<SUP>"</SUP> )} > β.</CENTER> A necessary and sufficient condition for a function to be in the class A<SUB>γ</SUB><SUP>λ, μ, ν</SUP>(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