We introduce and study a class of starlike functions defined by

$$\begin{aligned} {\mathscr {S}}^*_\wp :=\left\{ f\in {\mathcal {A}}: \frac{zf'(z)}{f(z)}\prec 1+ze^z=:\wp (z)\right\} , \end{aligned}$$

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 \( {\mathscr {S}}^*_\wp \).

中文翻译：

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心形区域和星形函数

我们介绍并研究由定义的一类星形函数

$$ \ begin {aligned} {\ mathscr {S}} ^ * _ \ wp：= \ left \ {f \ in {\ mathcal {A}}：\ frac {zf'（z）} {f（z） } \ prec 1 + ze ^ z =：\ wp（z）\ right \}，\ end {aligned} $$

其中\（\ wp \）将单位磁盘映射到心形区域。我们找到\（\ wp（z）\）的凸半径，并建立类\（{\ mathscr {S}} ^ * _ \ wp \）与一些知名类之间的包含关系。此外，我们得出类\（{\ mathscr {S}} ^ * _ \ wp \）的尖锐半径常数和与系数相关的结果。