A new differential subordination technique for a subclass of starlike functions

Abstract: In the present investigation, we employ a new technique to find several first and second order differential subordination implications involving the following starlike class associated with a bean shaped domain: \begin{equation*} \mathcal{S}^*_{\mathfrak{B}}:=\left{f\in\mathcal{S}:\dfrac{zf'(z)}{f(z)}\prec\sqrt{1+\tanh{z}}=:\mathfrak{B}(z)\right}. \end{equation*} Also, we give several applications stemming from our derived results.