A new differential subordination technique for a subclass of starlike functions
S. Sivaprasad Kumar, Pooja Yadav
TL;DR
The paper addresses differential-subordination problems for the bean-shaped starlike subclass $\mathcal{S}^*_{\mathfrak{B}}$, defined by $\dfrac{zf'(z)}{f(z)}\prec\mathfrak{B}(z)$ with $\mathfrak{B}(z)=\sqrt{1+\tanh z}$, by developing an admissible-function framework based on $q(z)=\mathfrak{B}(z)$ and the class $\Psi(\Omega, q)$. It introduces a new technique, supported by lower-bound lemmas, to deduce that subordinations of expressions like $(p(z))^\delta+\beta(zp'(z))^n$ (and variants) to $\mathfrak{B}$ force $p(z)\prec\mathfrak{B}(z)$, thereby yielding membership in $\mathcal{S}^*_{\mathfrak{B}}$ or $p(z)\prec\mathfrak{B}(z)$. The authors present extensive first- and second-order differential-subordination results for multiple nonlinear forms and target functions (e.g., $\mathfrak{B}$, linear-fractional $\dfrac{1+Az}{1+Bz}$, and $\sqrt{1+z}$), with explicit parameter bounds. These contributions extend the differential-subordination toolbox to the bean-domain and offer a flexible framework for analyzing subordination phenomena in related starlike subclasses.
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.