Star-likeness associated with the exponential function

Abstract

Given a domain Ω in the complex plane C and a univalent function q defined in an open unit disk D withnice boundary behaviour, Miller and Mocanu studied the class of admissible functions (Ω; q) so that the differentialsubordination (p(z); zp′(z); z2p′′(z); z) ≺ h(z) implies p(z) ≺ q(z) , where p is an analytic function in D with p(0) = 1, : C3 D ! C and Ω = h(D) . This paper investigates the properties of this class for q(z) = ez . As application, severalsufficient conditions for normalized analytic functions f to be in the subclass of star-like functions associated with theexponential function are obtained.