Toeplitz Determinants in One and Higher Dimensions

TL;DR

This paper establishes sharp bounds for Toeplitz determinants associated with holomorphic functions in one and higher dimensions, extending classical results to complex Banach spaces and polydiscs.

Contribution

It introduces new bounds for Toeplitz determinants for classes of holomorphic functions in multiple dimensions, expanding the scope of classical univalent function theory.

Findings

Derived sharp bounds for Toeplitz determinants in the unit disk.

Extended bounds to functions on the unit ball in Banach spaces.

Generalized results to functions on the polydisc in complex space.

Abstract

In this study, we derive the sharp bounds of certain Toeplitz determinants whose entries are the coefficients of holomorphic functions belonging to a class defined on the unit disk $\mathbb{U}$. Further, these results are extended to a class of holomorphic functions on the unit ball in a complex Banach space and on the unit polydisc in $\mathbb{C}^n$. The obtained results provide the bounds of Toeplitz determinants for various subclasses of normalized univalent functions in higher dimensions.