  In this paper, we introduce and explore a new class of starlike functions
denoted by $\mathcal{S}^*_{\mathfrak{B}}$, defined as follows:
  $$\mathcal{S}^*_{\mathfrak{B}}=\{f\in \mathcal{A}:zf'(z)/f(z)\prec
\sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\}.$$
  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
$\mathcal{S}^*_{\mathfrak{B}}$. We derive sharp conditions under which
$\psi(p)\prec\sqrt{1+\tanh(z)}$ implies $p(z)\prec ((1+A z)/(1+B z))^\gamma$,
where $\psi(p)$ is defined as:
  \begin{equation*} (1-\alpha)p(z)+\alpha p^2(z)+\beta
\frac{zp'(z)}{p^k(z)}\quad \text{and}\quad (p(z))^\delta+\beta
\frac{zp'(z)}{(p(z))^k}. \end{equation*} Additionally, we establish inclusion
relations involving $\mathcal{S}^*_{\mathfrak{B}}$ and derive precise estimates
for the sharp radii constants of $\mathcal{S}^*_{\mathfrak{B}}$.
