Schwarzian Norm Estimate for Functions in Generalized Robertson Class

TL;DR

This paper derives sharp estimates for the pre-Schwarzian and Schwarzian norms of functions within a generalized Robertson class characterized by specific real part conditions involving parameters lpha and eta.

Contribution

It provides the first sharp bounds for the Schwarzian and pre-Schwarzian norms for this generalized class of analytic functions.

Findings

Sharp estimates for Schwarzian norms are obtained.

Pre-Schwarzian norms are explicitly bounded.

Results extend known bounds to a broader class of functions.

Abstract

Let $\mathcal{A}$ be the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalized conditions $f(0)=0$, $f'(0)=1$. For $-\pi/2<\alpha<\pi/2$ and $0\le \beta<1$, let $\mathcal{S}_{\alpha}(\beta)$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $${\rm Re\,} \left\{e^{i\alpha}\left(1+\frac{zf''(z)}{f'(z)}\right)\right\}>\beta\cos{\alpha}\quad\text{for}~z\in\mathbb{D}.$$ In the present study, we will compute the sharp estimate of the pre-Schwarzian and Schwarzian norms for functions in $\mathcal{S}_{\alpha}(\beta)$.