An extremal problem for univalent functions

TL;DR

This paper provides precise bounds on a complex expression involving univalent functions, specifically targeting subclasses of normalized univalent functions within the unit disk, with the results being sharp and optimal.

Contribution

It introduces sharp estimates for a specific logarithmic and argument-based expression for subclasses of normalized univalent functions, advancing the understanding of their extremal properties.

Findings

Sharp bounds for (z)/z and its argument for univalent functions

Optimal estimates achieved for subclasses of normalized univalent functions

Enhanced understanding of extremal problems in geometric function theory

Abstract

For a real constant $b,$ we give sharp estimates of $\log|f(z)/z|+b\arg[f(z)/z]$ for subclasses of normalized univalent functions $f$ on the unit disk.