In this paper, we introduce and explore a new class of starlike functions, denoted as \({\mathcal {S}}^*_{{\mathfrak {B}}}\) , comprising normalized univalent analytic functions f that satisfy the condition \(zf'(z)/f(z)\prec \sqrt{1+\tanh {z}}=:{\mathfrak {B}}(z)\) , where \({\mathfrak {B}}(z)\) denotes a mapping from the unit disk onto a bean-shaped domain. Our investigation is focused on elucidating the characteristic properties of both \({\mathfrak {B}}(z)\) and the functions within \({\mathcal {S}}^*_{{\mathfrak {B}}}\) . We establish precise conditions under which the subordination \(\psi (p(z),zp'(z);z)\prec \sqrt{1+\tanh (z)}\) implies \(p(z)\prec ((1+A z)/(1+B z))^\gamma \) , where \(\psi (p(z),zp'(z);z)\) takes the form either of \((1-\alpha )p(z)+\alpha p^2(z)+\beta \frac{zp'(z)}{p^k(z)}\) or \((p(z))^\delta +\beta \frac{zp'(z)}{(p(z))^k}\) . Furthermore, we establish inclusion relations for \({\mathcal {S}}^*_{{\mathfrak {B}}}\) and provide estimations for sharp radii constants pertaining to \({\mathcal {S}}^*_{{\mathfrak {B}}}\) .
