On a subfamily of starlike functions related to Hyperbolic Cosine function

Abstract

We introduce and study a new Ma-Minda subclass of starlike functions ${\mathcal{S}}_{\varrho }^{\ast },$ defined as $${\mathcal{S}}_{\varrho }^{\ast }:=\left\{f\in \mathcal{A}:\frac{z{f}^{\prime }(z)}{f(z)}\prec \cosh \sqrt{z}=:\varrho (z),z\in \mathbb{D}\right\},$$ associated with an analytic univalent function $\cosh \sqrt{z},$ where we choose the branch of the square root function so that $\cosh \sqrt{z}=1+z/2!+z^2/4!+\cdots .$ We establish certain inclusion relations for ${\mathcal{S}}_{\varrho }^{\ast }$ and deduce sharp ${\mathcal{S}}_{\varrho }^{\ast }-$radii for certain subclasses of analytic functions.