Geometric properties of a domain with cusps

TL;DR

This paper investigates the geometric properties of a specific class of starlike functions associated with epicycloid domains with cusps, including inclusion relations, coefficient bounds, and radii problems.

Contribution

It introduces and analyzes the class al S_{nL} of starlike functions mapped onto epicycloid domains, providing new inclusion relations, coefficient bounds, and radii results.

Findings

Established inclusion relations with other starlike subclasses

Computed bounds on initial coefficients of functions in the class

Solved various radii problems for the class al S_{nL}

Abstract

For $n\geq 4$ (even), the function $\varphi_{n\mathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the unit disk $\mathbb{D}$ onto a domain bounded by an epicycloid with $n-1$ cusps. In this paper, the class $\mathcal{S}^*_{n\mathcal{L}} = \mathcal{S}^*(\varphi_{n\mathcal{L}})$ is studied and various inclusion relations are established with other subclasses of starlike functions. The bounds on initial coefficients is also computed. Various radii problems are also solved for the class $\mathcal{S}^*_{n\mathcal{L}}$.