Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

TL;DR

This paper introduces a new class of starlike functions related to a balloon-shaped domain and derives bounds for various determinants involving their coefficients, advancing geometric function theory applications.

Contribution

It defines a novel class of starlike functions linked to a specific geometric domain and establishes coefficient bounds for associated determinants.

Findings

Bounds for second order Hankel determinants

Bounds for second order Toeplitz determinants

Results on logarithmic coefficients and inverse functions

Abstract

Recent advances in image and signal processing have drawn on geometric function theory, particularly coefficient estimate problems. Motivated by their significance, we introduce a class of starlike functions related to a balloon-shaped domain \[ \mathcal{S}^*_{\mathcal{B}}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} := B(z); \; z \in \mathbb{D} \right\}, \] where $B(z)$ maps the unit disk $\mathbb{D}$ onto a balloon-shaped domain. This work establishes bounds for the second order Hankel determinants and second order Toeplitz determinants involving the initial coefficients, the logarithmic coefficients and the logarithmic coefficients of the inverse function for $f \in \mathcal{S}^*_{\mathcal{B}}$