Geometric properties of a domain with cusps

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 $\mathcal{S}^*_{n\mathcal{L}}$.