Radii problems for Ma-Minda Starlikeness

TL;DR

This paper investigates radii problems within the Ma-Minda class of univalent starlike functions, deriving radii for special functions, extremal functions for certain maximization problems, and results on convolution and radius estimates.

Contribution

It introduces new radius results and extremal function characterizations for the Ma-Minda class, extending classical problems in geometric function theory.

Findings

Derived $ ext{S}^*( ext{ extpsi})$-radii for special functions.

Identified extremal functions for maximization problems involving $ ext{S}^*( ext{ extpsi})$.

Established convolution and radius estimates for starlike images.

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.