Some Differential Subordination Results for a Class of Starlike Functions

In this paper, first order differential subordination implication results are derived for $\mathcal{S}^{*}_{\varrho},$ a subclass of starlike functions, defined as $$\mathcal{S}^{*}_{\varrho}=\left\{f\in\mathcal{A}:\frac{zf'(z)}{f(z)}\prec \varrho(z):=\cosh \sqrt{z} ,z\in\mathbb{D}\right\},$$ where we choose the branch of the square root function so that $\cosh\sqrt{z}=1+z/2!+z^{2}/{4!}+\cdots.$ Further we deduce Briot-Bouquet differential subordination results along with some examples.Comment: arXiv admin note: substantial text overlap with arXiv:2201.0581