Pre-Schwarzian and Schwarzian norm Estimates for Robertson class

TL;DR

This paper investigates the properties of a subclass of analytic functions defined by a real part condition, providing distortion, growth, and norm estimates, including sharp bounds for the Schwarzian norm based on initial derivatives.

Contribution

It offers a new characterization of Robertson functions and derives explicit distortion, growth, and sharp norm bounds for the subclass lpha functions.

Findings

Derived distortion and growth theorems for lpha functions.

Established sharp upper bounds for Schwarzian norms.

Provided a characterization linking initial derivatives to norm bounds.

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 $-\pi/2<\alpha<\pi/2$, let $\mathcal{S}_{\alpha}$ be the subclass of $\mathcal{A}$ consisting of functions $f$ that satisfy the relation $\mathrm{Re}\{e^{i\alpha}\left(1+zf^{\prime\prime}(z)/f^{\prime}(z)\right)\}>0$ 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 $\mathcal{S}_{\alpha}$. In addition, a sharp upper bound of the Schwarzian norm for the subclass is given in terms of the value $f^{\prime \prime}(0)$.