A class of univalent functions defined by a differential inequality

Let A be the class of analytic functions in the unit disk D with the normalization f (0) = f′ (0) − 1 = 0. For λ > 0, denote by M (λ) the class of functions f A which satisfy the condition |z² (zf (z) ) ''+ f' (z) (zf (z) ) ^2-1 |, z D. We show that functions in M (1) are univalent in D and we present one parameter family of functions in M (1) that are also starlike in D. In addition to certain inclusion results, we also present characterization formula, necessary and sufficient coefficient conditions for functions in M (λ), and a radius property of M (1).