On Coefficient problems for \textbf{$S^*_{\rho}$}

TL;DR

This paper investigates coefficient bounds for functions in the starlike class \\mathcal{S}^*_ ho, focusing on logarithmic and inverse logarithmic coefficients, and establishes bounds for Hankel and Toeplitz determinants related to these coefficients.

Contribution

It introduces new bounds for Hankel and Toeplitz determinants for functions in the class \\mathcal{S}^*_ ho, based on their logarithmic coefficients, extending existing coefficient problems.

Findings

Bounds for second Hankel determinants established.

Bounds for Toeplitz determinants determined.

Results specific to the petal-shaped domain mapped by \\rho(z).

Abstract

Logarithmic and inverse logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the class of starlike functions \(\mathcal{S}^*_\rho\), defined as \[ \mathcal{S}^*_\rho = \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \rho(z), \; z \in \mathbb{D} \right\}, \] where \(\rho(z) := 1 + \sinh^{-1}(z)\), which maps the unit disk \(\mathbb{D}\) onto a petal-shaped domain. This investigation aims to establish bounds for the second Hankel and Toeplitz determinants, with their entries determined by the logarithmic coefficients of \(f\) and its inverse \(f^{-1}\), for functions \(f \in \mathcal{S}^*_\rho\).