Certain geometric properties of starlike functions related to a limacon domain

Abstract

Let $\mathcal{ST}_{lim}$ be the family of starlike functions associated with $\varphi_{\scriptscriptstyle{lim}}(z)=1+\sqrt{2}z+z^2/2$, the function which sends the open unit disc $\mathbb{D}$ univalently onto a region bounded by the dimpled limacon curve $\left(9u^2+9v^2-18u+5\right)^2-16\left(9u^2+9v^2-6u+1\right)=0$. In this paper, we study certain geometrical properties along with some inclusion results related to the function class $\mathcal{ST}_{lim}$. Further, given some analytic function $\mathcal{P}(z)$ having nice geometrical properties, we obtain sharp estimates on the real $\beta$ such that the first-order differential subordination