On Starlike Functions Associated with a Bean Shaped Domain

Abstract

In this paper, we introduce and explore a new class of starlike functions denoted by ${\mathcal{S}}_{\mathfrak{B}}^{\ast }$, defined as follows: $${\mathcal{S}}_{\mathfrak{B}}^{\ast }=\{f\in \mathcal{A}:z{f}^{\prime }(z)/f(z)\prec \sqrt{1+\tanh z}=:\mathfrak{B}(z)\}.$$ Here, $\mathfrak{B}(z)$ represents a mapping from the unit disk onto a bean-shaped domain. Our study focuses on understanding the characteristic properties of both $\mathfrak{B}(z)$ and the functions in ${\mathcal{S}}_{\mathfrak{B}}^{\ast }$. We derive sharp conditions under which $\ddot{I}ˆ(p)\prec \sqrt{1+\tanh (z)}$ implies $p(z)\prec ((1+Az)/(1+Bz){)}^3$, where $\ddot{I}ˆ(p)$ is defined as: \begin{equation*} (1-α)p(z)+αp^2(z)+β\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^δ+β\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion relations involving ${\mathcal{S}}_{\mathfrak{B}}^{\ast }$ and derive precise estimates for the sharp radii constants of ${\mathcal{S}}_{\mathfrak{B}}^{\ast }$.