Second-Order Toeplitz Determinant for Quasi-Convex Mappings

TL;DR

This paper derives sharp bounds for the second-order Toeplitz determinant associated with convex functions in the unit disk, extending results to complex Banach spaces and polydisks, impacting the study of quasi-convex mappings.

Contribution

It introduces new sharp estimates for the second-order Toeplitz determinant for convex functions, extending these bounds to broader classes of holomorphic mappings.

Findings

Sharp bounds for second-order Toeplitz determinants for convex functions

Extension of estimates to holomorphic mappings in Banach spaces

Bounds applicable to quasi-convex mappings of type B

Abstract

This paper presents sharp estimates for the second-order Toeplitz determinant whose entries are the coefficients of convex functions defined on the unit disk in $\mathbb{C}$. These estimates are further extended to a subclass of holomorphic mappings defined on the unit ball in a complex Banach space and on the unit polydisk in $\mathbb{C}^n$, which, as special cases, yield bounds for the classes of quasi-convex mappings of type $B$.