On a subfamily of starlike functions related to hyperbolic cosine function

We introduce and study a new Ma-Minda subclass of starlike functions S-?(& lowast;), defined as S-?( )& lowast;:= { f is an element of A :(f (z))/ (zf ' (z) )? cosh root z =: ? (z), z is an element of D }, associated with an analytic univalent function cosh root z, where we choose the branch of the square root function so that cosh root z = 1 + z /2! + z(2 )/ 4! + ? We establish certain inclusion relations for S-?(& lowast;) and deduce sharp S-?(& lowast;)-radii for certain subclasses of analytic functions.