Pre-Schwarzian and Schwarzian norm Estimates for Robertson class
Molla Basir Ahamed, Rajesh Hossain, Xiaoyuan Wang
TL;DR
The paper addresses estimating the pre-Schwarzian and Schwarzian norms for Robertson-class functions $\mathcal{S}_α$, establishing a differential-subordination framework to obtain sharp distortion, growth, and norm bounds. It derives the key inequality $\Re\{e^{iα}(1+\frac{zf''(z)}{f'(z)})\}\ge 1-\cos α+\frac{1-|z|^2}{4\cos α}|\frac{f''(z)}{f'(z)}|^2$ and a geometric bound $(1-|z|^2)|\frac{f''(z)}{f'(z)}-2\cos α \bar z|\le 2\cos α$, yielding distortion and growth theorems. The main norm estimates are $||Pf||\le 2\cos α$ and $||Sf||\le 2\cos α(2-\cos α)$, with a refined Schwarzian bound in terms of $\gamma=\frac{|f''(0)|}{2\cos α}$, and the extremal function $f_α(z)=\int_{0}^{z}(1-\xi^2)^{-\cos α}d\xi$ attains sharpness; the results specialize to the classical convex case when $α=0$. These findings advance univalence and quasiconformal extension insights for Robertson-type classes and connect to known convex-function bounds, providing a coherent α-dependent framework for Schwarzian/pre-Schwarzian analysis.
Abstract
Let $\mathcal{A}$ denote the class of analytic functions $f$ on the unit disk $\mathbb{D}=\{z\in\mathbb{C} : |z|<1\}$, normalized by $f(0)=0$ and $f^{\prime}(0)=1$. For $-π/2<α<π/2$, let $\mathcal{S}_α$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{iα}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ for $z\in\mathbb{D}$. In this paper, we first give an equivalent characterization for a subclass of Robertson functions; then we present the distortion and growth theorems and obtain the pre-Schwarzian and Schwarzian norms for the subclass $\mathcal{S}_α$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$.