Pre-Schwarzian norm estimation for functions in the Ma-Minda-type starlike and convex classes

TL;DR

This paper derives sharp estimates for the pre-Schwarzian norm of functions within specific Ma-Minda-type starlike and convex classes, expanding understanding of their geometric properties based on various defining functions.

Contribution

The paper provides the first sharp bounds for the pre-Schwarzian norm in Ma-Minda-type classes with multiple parameterized defining functions.

Findings

Sharp estimates for pre-Schwarzian norms established

Results cover various parameter ranges for defining functions

Enhances understanding of geometric function theory in these classes

Abstract

In this paper, we establish the sharp estimates of the pre-Schwarzian norm of functions $f$ in the Ma-Minda type starlike and convex classes $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$, respectively, whenever $\varphi(z)=3/\left(3+(\alpha-3)z-\alpha z^2\right)$ with $-3<\alpha\leq 1$, $\varphi(z)=(1+z)(1-s z)$ with $-1/3\leq s\leq 1/3$ and $\varphi(z)=1+z/\left((1-z) (1+\alpha z)\right)$ with $0\leq \alpha\leq 1/2$.