A class of univalent functions defined by using Hadamard product

Abstract

In this paper we introduce the class $L_{\alpha}^*(\lambda,\beta)$ of functions defined by $f*S_{\alpha}(z)$ of $f(z)$ and $S_{\alpha}=\dfrac z{(1-z)^{2(1-\alpha)}}$. We determine coefficient estimates, closure theorems, distortion theorems and radii of close-to-convexity, starlikeness and convexity. Also we find integral operators and some results for Hadamard products of functions in the class $L_{\alpha}^*(\lambda,\beta)$. Finally, in terms of the operators of fractional calculus, we derive several sharp results depicting the growth and distortion properties of functions belonging to the class $L_{\alpha}^*(\lambda,\beta)$.