Toeplitz Determinants for Inverse Functions and their Logarithmic Coefficients Associated with Ma-Minda Classes
Surya Giri
TL;DR
This work addresses sharp bounds for Toeplitz determinants formed from the coefficients of inverse functions and from the logarithmic coefficients of inverse functions for Ma-Minda subclasses $\mathcal{S}^*(\varphi)$ and $\mathcal{C}(\varphi)$. It uses subordination to $\varphi$ and the Schwarz-class framework to derive exact bounds for $|T_{2,1}(f^{-1})|$, $|T_{2,2}(f^{-1})|$, $|T_{2,1}(F_{f^{-1}})|$, and $|T_{2,2}(F_{f^{-1}})|$, with extremal functions $f_\varphi$ and $h_\varphi$ confirming sharpness. The results cover general Ma-Minda classes and yield extensive, explicit corollaries for many known starlike and convex subclasses, including $\mathcal{S}^*[A,B]$, $\mathcal{C}[A,B]$, $\mathcal{S}^*(\alpha)$, $\mathcal{C}(\alpha)$, etc. The findings enhance understanding of inverse-coefficient structures and provide tools for applications in geometric function theory and related areas.
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.