A Cardioid Domain and Starlike Functions

TL;DR

This paper introduces a new class of starlike functions associated with a cardioid domain, analyzes their geometric properties, and establishes inclusion relations, radius constants, and coefficient bounds.

Contribution

It defines a novel class of starlike functions based on a cardioid mapping and investigates their geometric and coefficient properties.

Findings

Determined the radius of convexity for the cardioid domain.

Established inclusion relations with known starlike classes.

Derived sharp radius constants and coefficient bounds for the new class.

Abstract

We introduce and study a class of starlike functions defined by \begin{equation*} \mathscr{S}^*_\wp:=\left\{f\in\mathcal{A}: \frac{zf'(z)}{f(z)}\prec 1+ze^z=:\wp(z)\right\}, \end{equation*} where $\wp$ maps the unit disk onto a cardioid domain. We find the radius of convexity of $\wp(z)$ and establish the inclusion relations between the class $ \mathscr{S}^*_\wp$ and some well-known classes. Further we derive sharp radius constants and coefficient related results for the class $ \mathscr{S}^*_\wp$.