A cardioid domain and starlike functions

We introduce and study a class of starlike functions defined by S℘∗:=f∈A:zf′(z)f(z)≺1+zez=:℘(z),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {S}}^*_\wp :=\left\{ f\in {\mathcal {A}}: \frac{zf'(z)}{f(z)}\prec 1+ze^z=:\wp (z)\right\} , \end{aligned}$$\end{document}where ℘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\wp $$\end{document} maps the unit disk onto a cardioid domain. We find the radius of convexity of ℘(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\wp (z)$$\end{document} and establish the inclusion relations between the class S℘∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathscr {S}}^*_\wp $$\end{document} and some well-known classes. Further we derive sharp radius constants and coefficient related results for the class S℘∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathscr {S}}^*_\wp $$\end{document}.