Schwarzian norm estimates for some classes of analytic functions

TL;DR

This paper provides sharp estimates of the Schwarzian norm for certain classes of normalized analytic functions in the unit disk, and establishes two-point distortion theorems for these classes.

Contribution

It introduces sharp Schwarzian norm bounds for classes al G(eta) and al F(\u03b1), and proves two-point distortion theorems for these classes.

Findings

Sharp Schwarzian norm estimates for al G(eta) and al F(1/2) classes.

Two-point distortion theorems established for these classes.

Results improve understanding of geometric properties of these function classes.

Abstract

Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f'(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(\beta)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+\beta/2\}$, where $\beta>0$ and $\mathcal{F}(\alpha)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>\alpha\}$, where $-1/2\le \alpha\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(\beta)$ and $\mathcal{F}(\alpha)$.