Radius of starlikeness of $\mathcal{S}\ast \mathcal{S}t(\alpha )$

Abstract

Let $\mathcal{S}$ be the set of all analytic univalent functions $f$ defined in the open unit disc $\mathbb{D}$, with $f(0)=0=f^{\prime }(0)-1$. For $\alpha \in [0,1)$, let $\mathcal{S}t(\alpha )$ be the set of all starlike functions of order $\alpha$ in $\mathcal{S}$. In this article, by applying duality technique we obtain the radius of a disc that is mapped onto a starlike domain with respect to the origin by the functions in the set $\mathcal{S}\ast \mathcal{S}t(\alpha ):=\{f\ast g:f\in \mathcal{S},g\in \mathcal{S}t(\alpha )\}$. Here, `$\ast$' denotes the convolution (or Hadamard product) of two analytic functions in $\mathbb{D}$.

Submitted for publication