Applications of fractional derivatives for Alexander integral operator

Abstract

Let $$T_n$$ be the class of functions $$f(z)=z+a_{n+1}z^{n+1}+a_{n+2}z^{n+2}+\ldots $$ that are analytic in the closed unit disc $$\mathbb {{\overline{U}}}.$$ With m different boundary points $$z_{s}, (s=1,2,\ldots , m),$$ we consider $$\alpha _{m}\in e^{i\beta }A_{j+\lambda }f({\mathbb {U}}),$$ here $$A_{j+\lambda }$$ is given by using fractional derivatives $$D_{j+\lambda }f(z)$$ for $$f(z)\in T_n.$$ Using $$A_{j+\lambda },$$ we introduce a subclass $$P_{n}(\alpha _{m}, \beta , \rho ; j, \lambda ) $$ of $$T_n.$$ The main goal of our paper is to discuss some interesting results of f(z) in the class $$P_{n}(\alpha _{m}, \beta , \rho ; j, \lambda ).$$