  We introduce and study a new Ma-Minda subclass of starlike functions
$\mathcal{S}^*_{\varrho},$ defined as
$$\mathcal{S}^{*}_{\varrho}:=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)} \prec
\cosh \sqrt{z}=:\varrho(z), z\in\mathbb{D} \right\},$$ 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 $\mathcal{S}^{*}_{\varrho}$ and
deduce sharp $\mathcal{S}^{*}_{\varrho}-$radii for certain subclasses of
analytic functions.

Complex Variables