Sharp radius of concavity for certain classes of analytic functions

TL;DR

This paper determines the largest radius within which certain subclasses of normalized analytic functions on the unit disk are concave, providing optimal bounds for these radii across various classes.

Contribution

It introduces the sharp radius of concavity for multiple subclasses of analytic functions, extending existing results with best possible bounds.

Findings

Established the sharp radius of concavity for subclasses like , , , and .

Provided best possible radii bounds for these classes.

Extended the analysis to various classes of analytic functions on the unit disk.

Abstract

Let $\mathcal{A}$ be the class of all analytic functions $f$ defined on the open unit disk $\mathbb{D}$ with the normalization $f(0)=0=f^{\prime}(0)-1$. This paper examines the radius of concavity for various subclasses of $\mathcal{A}$, namely $\mathcal{S}_0^{(n)}$, $\mathcal{K(\alpha,\beta)}$, $\mathcal{\tilde{S^*}(\beta)}$, and $\mathcal{S}^*(\alpha)$. It also presents results for various classes of analytic functions on the unit disk. All the radii are best possible.