Toeplitz determinants whose elements are the coefficients of univalent functions

TL;DR

This paper establishes sharp bounds for Toeplitz determinants formed from Taylor coefficients of univalent functions and explores related problems for typically real functions, advancing understanding of coefficient inequalities in complex analysis.

Contribution

It provides the first sharp estimates for Toeplitz determinants of univalent functions and extends the analysis to certain subclasses and real functions.

Findings

Sharp bounds for Toeplitz determinants of univalent functions

Results for specific subclasses of univalent functions

Analysis of Toeplitz determinants for typically real functions

Abstract

Let $\mathcal{S}$ denote the class of analytic and univalent functions in $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$. In this paper, we determine sharp estimates for the Toeplitz determinants whose elements are the Taylor coefficients of functions in $\mathcal{S}$ and its certain subclasses. We also discuss similar problems for typically real functions.