Schwarzian derivative for convex mappings of order alpha

TL;DR

This paper derives sharp bounds for the Schwarzian derivative norm, distortion, and growth of convex mappings of order alpha, extending previous results and providing precise estimates based on initial derivatives.

Contribution

It introduces new sharp bounds for the Schwarzian derivative norm and related properties for convex mappings of order alpha, generalizing prior work by Suita and Yamashita.

Findings

Sharp bounds for Schwarzian derivative norm in terms of f''(0)

Sharp distortion and growth bounds for convex mappings of order alpha

Generalization of previous results by Suita and Yamashita

Abstract

The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order $alpha$ in terms of the value of $f''(0)$, in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita and Yamashita for this particular case.