Sharp norm estimate of Schwarzian derivative for a class of convex functions

TL;DR

This paper derives a precise upper bound for the Schwarzian derivative norm for convex functions, including subclasses like strongly convex and uniformly convex functions, enhancing understanding of their geometric properties.

Contribution

It provides the first sharp norm estimates of the Schwarzian derivative for convex functions and their subclasses, extending previous bounds.

Findings

Sharp norm estimate for convex functions' Schwarzian derivative

Sharp bounds for strongly convex functions of order α

Sharp bounds for uniformly convex functions

Abstract

We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, International Press Inc., 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order $\alpha,~0<\alpha<1,$ and for uniformly convex functions.