Radii problems for Ma-Minda Starlikeness

Abstract

For the standard Ma-Minda class ${\mathcal{S}}^{\ast }(\psi )$ of univalent starlike functions, we derive ${\mathcal{S}}^{\ast }(\psi )$-radii for some well-known special functions. In addition, we obtain the set of extremal functions for the classical problem $$\underset{f\in {\mathcal{S}}^{\ast }(\psi )}{\max }\left|\Phi \left(\log (f(z)/z)\right)\right|\text{or}\underset{f\in {\mathcal{S}}^{\ast }(\psi )}{\max }\Re \Phi \left(\log (f(z)/z)\right),$$ where $\Phi$ is a non-constant entire function. Moreover, we prove certain results on convolution and radius estimates for the case when $\psi (\mathbb{D})$ is starlike.