Univalence criteria and analogs of the John constant

TL;DR

This paper establishes bounds on certain constants related to the function p(z) = zf'(z)/f(z) that ensure the univalence of functions analytic in the unit disk, extending classical criteria with new bounds.

Contribution

It provides new lower and upper bounds for constants ensuring univalence based on properties of p(z), generalizing univalence criteria involving the John constant.

Findings

Derived bounds for  and  constants

Established conditions linking p(z) bounds to univalence

Extended classical univalence criteria

Abstract

Let $p(z)=zf'(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0, f'(0)=1.$ We will provide lower and upper bounds for the best constants $\delta_0$ and $\delta_1$ such that the conditions $e^{-\delta_0/2}<|p(z)|<e^{\delta_0/2}$ and $|p(w)/p(z)|<e^{\delta_1}$ for $|z|,|w|<1$ respectively imply univalence of $f$ on the unit disk.