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 $f(0)=0=f'(0)-1$. 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 $\mathbb{D}$.