A new subclass R ( α ) , 0 ≤ α ≤ 1 , of the class S t ( 1 / 2 ) the class of starlike functions of order 1 / 2 is introduced and it is shown that R ( α ) is closed with respect to the Hadamard product of analytic functions. Some sufficient conditions for the normalized regular functions to be univalent in the unit disk E are given.