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

Abstract

Abstract In this paper, we study a class of univalent functions f as defined by making use of the generalized Ruscheweyh derivatives involving a general fractional derivative operator, satisfying Re { z ( J 1 λ , μ f ( z ) ) ' ( 1 - γ ) J 1 λ , μ f ( z ) + γ z 2 ( J 1 λ , μ f ( z ) ) ' ' } > β . {\mathop{\rm Re}\nolimits} \left\{{{{z\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)'} \over {\left({1 - \gamma} \right){\bf{J}}_1^{\lambda,\mu}f\left(z \right) + \gamma {z^2}\left({{\bf{J}}_1^{\lambda,\mu}f\left(z \right)} \right)''}}} \right\} > \beta. A necessary and sufficient condition for a function to be in the class A γ λ , μ , ν ( n , β ) A_\gamma ^{\lambda,\mu,\nu}\left({n,\beta} \right) is obtained. Also, our paper includes linear combination, integral operators and we introduce the subclass A γ , c m λ , μ , ν ( 1 , β ) A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) consisting of functions with negative and fixed finitely many coefficients. We study some interesting properties of A γ , c m λ , μ , ν ( 1 , β ) A_{\gamma,{c_m}}^{\lambda,\mu,\nu}\left({1,\beta} \right) .