Schwarzian Norm Estimate for Functions in Robertson Class

TL;DR

This paper establishes the exact bounds for the Schwarzian and pre-Schwarzian norms of functions within a specific subclass of analytic functions characterized by a real part condition involving a parameter alpha.

Contribution

It provides the first sharp estimates of these norms for the Robertson class of functions, extending the understanding of their geometric properties.

Findings

Sharp bounds for Schwarzian norms are derived.

Sharp bounds for pre-Schwarzian norms are established.

Results apply to functions with a specified real part condition.

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 $f(0)=0$, $f'(0)=1$. For $-\pi/2<\alpha<\pi/2$, let $\mathcal{S}_{\alpha}$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation ${\rm Re\,} \{e^{i\alpha}(1+zf''(z)/f'(z))\}>0$ for $z\in\mathbb{D}$. In the present article, we determine the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{S}_{\alpha}$.