Starlike Functions Associated with a Non-Convex Domain

TL;DR

This paper introduces a new class of starlike functions linked to a non-convex domain, providing growth, distortion, and coefficient bounds, along with determinant estimates and inclusion relations, advancing geometric function theory.

Contribution

It defines a novel class of starlike functions associated with a specific non-convex domain and derives fundamental properties and bounds, enriching the understanding of such functions.

Findings

Established growth and distortion theorems.

Derived sharp coefficient bounds.

Provided estimates for Hankel and Toeplitz determinants.

Abstract

We introduce and study a class of starlike functions associated with the non-convex domain \[ \mathcal{S}^*_{nc} = \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1+z}{\cos{z}} =: \varphi_{nc}(z), \;\; z \in \mathbb{D} \right\}. \] Key results include the growth and distortion theorems, initial coefficient bounds, and the sharp estimates for third-order Hankel and Hermitian-Toeplitz determinants. We also examine inclusion relations, radius problems for certain subclasses, and subordination results. These findings enrich the theory of starlike functions associated with non-convex domains, offering new perspectives in geometric function theory.