Growth, Distortion, Pre-Schwarzian and Schwarzian norm estimates for Generalized Robertson class

TL;DR

This paper derives sharp bounds and distortion theorems for generalized Robertson class functions, focusing on their Schwarzian and pre-Schwarzian norms, and determines key radii related to their geometric properties.

Contribution

It provides new sharp bounds for Schwarzian and pre-Schwarzian norms, and establishes distortion, growth, and radius theorems for the generalized Robertson class.

Findings

Sharp bounds for Schwarzian and pre-Schwarzian norms derived.

Distortion and growth theorems established for the class.

Sharp radii of concavity and convexity determined.

Abstract

This paper investigates the geometric properties of functions within the generalized Robertson class which consists of alpha-starlike functions of order beta. The study's significance lies in providing a deeper understanding of the univalence and geometric behavior of these functions, which are fundamental in complex analysis and geometric function theory. The primary objective is to derive sharp bounds for the norms of the Schwarzian and pre-Schwarzian derivatives for functions in this class. These bounds are expressed in terms of the initial coefficient of the function, specifically focusing on the important case where this initial coefficient is zero. Additionally, the paper establishes sharp distortion and growth theorems for the functions belonging to this generalized class. Finally, the research addresses the radius problem for this function class by determining the sharp radius of concavity and the sharp radius of convexity.