Starlikeness Associated With The Exponential Function

Abstract

Given a domain $\Omega$ in the complex plane $\mathbb{C}$ and a univalent function $q$ defined in an open unit disk $\mathbb{D}$ with nice boundary behaviour, Miller and Mocanu studied the class of admissible functions $\Psi (\Omega ,q)$ so that the differential subordination $\psi (p(z),zp(z),z^2{p}^{\prime \prime }(z);z)\prec h(z)$ implies $p(z)\prec q(z)$ where $p$ is an analytic function in $\mathbb{D}$ with $p(0)=1$, $\psi :{\mathbb{C}}^3\times \mathbb{D}\to \mathbb{C}$ and $\Omega =h(\mathbb{D})$. This paper investigates the properties of this class for $q(z)=e^z$. As application, several sufficient conditions for normalized analytic functions $f$ to be in the subclass of starlike functions associated with the exponential function are obtained.