Generalized Toeplitz determinants for Starlike Mappings in Several Complex Variables

TL;DR

This paper derives sharp bounds for Toeplitz determinants related to starlike functions in complex analysis, extending classical results to higher dimensions and more general domains.

Contribution

It introduces generalized bounds for Toeplitz determinants for starlike mappings in multiple complex variables, broadening the scope of existing univariate results.

Findings

Established sharp bounds for second and third-order Toeplitz determinants.

Extended bounds to mappings in complex Banach spaces and bounded starlike domains.

Generalized several known bounds as special cases.

Abstract

This paper establishes sharp bounds for the second and third-order Toeplitz determinants associated with starlike functions $f$ in the unit disk such that $f(z)-z$ has a zero of order $k+1$ at $z=0$. These bounds are further extended to starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. The derived results generalize several known bounds as special cases.