$\mathcal{S}^{*}(\phi)$ and $\mathcal{C}(\phi)$-radii for some special functions

TL;DR

This paper determines the sharp radii for Ma-Minda classes of analytic functions, specifically for subclasses related to special functions, including the radii of strong starlikeness, expanding understanding of their geometric properties.

Contribution

It introduces sharp radii results for Ma-Minda classes of functions, particularly for classes $ ext{S}^*( ext{phi})$ and $ ext{C}( ext{phi})$, related to special functions and their geometric characteristics.

Findings

Sharp $ ext{S}^*( ext{phi})$-radii for certain special functions.

Sharp $ ext{C}( ext{phi})$-radii for specific subclasses.

Radius of strong starlikeness analyzed.

Abstract

In this paper, we consider the Ma-Minda classes of analytic functions $\mathcal{S}^{*}(\phi):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)}) \prec \phi(z) \}$ and $\mathcal{C}(\phi):= \{f\in \mathcal{A} : (1+{zf''(z)}/{f'(z)}) \prec \phi(z) \}$ defined on the unit disk $\mathbb{D}$ and show that the classes $\mathcal{S}^{*}(1+\alpha z)$ and $\mathcal{C}(1+\alpha z)$, $0<\alpha \leq 1$ solve the problem of finding the sharp $\mathcal{S}^{*}(\phi)$-radii and $\mathcal{C}(\phi)$-radii for some normalized special functions, whenever $\phi(-1)=1-\alpha$. Radius of strongly starlikeness is also considered.