Toeplitz Determinants for Inverse Functions and their Logarithmic Coefficients Associated with Ma-Minda Classes

TL;DR

This paper derives sharp bounds for Toeplitz determinants related to inverse functions and their logarithmic coefficients within Ma-Minda classes, unifying many subclasses of starlike and convex functions.

Contribution

It provides the first sharp bounds for Toeplitz determinants over inverse functions in Ma-Minda classes, extending known results to a broad class of univalent functions.

Findings

Established sharp bounds for Toeplitz determinants.

Results applicable to many subclasses of starlike and convex functions.

Unified approach covering various well-known classes.

Abstract

The classes of analytic univalent functions on the unit disk defined by $$ \mathcal{S}^*(\varphi)= \bigg\{ f \in \mathcal{A}: \frac{z f'(z)}{f(z)} \prec \varphi(z)\bigg\}$$ and $$ \mathcal{C}(\varphi)=\bigg\{ f \in \mathcal{A}: 1 + \frac{z f''(z)}{f'(z)} \prec \varphi(z)\bigg\} $$ generalize various subclasses of starlike and convex functions, respectively. In this paper, sharp bounds are established for certain Toeplitz determinants constructed over the coefficients and logarithmic coefficients of inverse functions belonging to $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$. Since these classes covers many well-known subclasses, the derived bounds are directly applicable to them as well.