A new differential subordination technique for a subclass of starlike functions

TL;DR

This paper introduces a novel differential subordination technique to analyze a specific subclass of starlike functions associated with a bean-shaped domain, providing new implications and applications in geometric function theory.

Contribution

The paper develops a new differential subordination method tailored for a subclass of starlike functions related to a bean-shaped domain, advancing the theoretical framework.

Findings

Derived several first and second order differential subordination implications.

Established new inclusion and growth results for the subclass.

Provided multiple applications of the main results.

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.