Radii problems for Ma-Minda Starlikeness

Abstract

For the standard Ma-Minda class $\mathcal{S}^{*}(\psi)$ of univalent starlike functions, we derive $\mathcal{S}^{*}(\psi)$-radii for some well-known special functions. In addition, we obtain the set of extremal functions for the classical problem $$\max_{f\in \mathcal{S}^{*}(\psi)}^{}\left|\Phi\left(\log{(f(z)/z)}\right)\right| \quad \text{or} \quad \max_{f\in \mathcal{S}^{*}(\psi)}^{}\Re\Phi\left(\log{(f(z)/z)}\right),$$ 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.