Second-Order Toeplitz Determinant for Quasi-Convex Mappings

TL;DR

This work addresses sharp bounds for the second-order Toeplitz determinant $|T_{2,3}(f)|$ for convex-type function classes via subordination to a biholomorphic map $\Psi$, and extends the framework to higher-dimensional holomorphic mappings on the unit ball and unit polydisk. The authors derive explicit, sharp inequalities by linking coefficient bounds to the derivatives of $\Psi$ and to structured Schwarz-function coefficients, culminating in a general result for $f\in \mathcal{C}(\Psi)$ and several important corollaries for classical convex classes. They then lift these one-variable results to several complex variables, proving sharp determinant-type bounds for mappings with $F(z)= z f(z)$ lying in $\mathcal{C}(\mathbb{B})$ or $\mathcal{C}_\alpha(\mathbb{B})$, and provide analogous bounds on the polydisk, with extremal mappings $F_\Psi$ achieving equality. The paper thus unifies and extends sharp Toeplitz-determinant bounds from one-variable convex mappings to higher-dimensional holomorphic mappings, offering concrete inequalities and extremals for quasi-convex mappings of type $B$ in multiple settings.

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 $B$.

Theorems & Definitions (17)

• Definition 1
• Definition
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Theorem 4
• proof
• Corollary 4.1
• Corollary 4.2