On a class of univalent functions

Let A be the class of analytic functions in the unit disk D with the normalization f ( 0 ) = f ′ ( 0 ) − 1 = 0 . Denote by N the class of functions f ∈ A which satisfy the condition | − z 3 ( z f ( z ) ) ‴ + f ′ ( z ) ( z f ( z ) ) 2 − 1 | ≤ 1 , z ∈ D . We show that functions in N are univalent in D but not necessarily starlike. Also, we present the characterization formula, necessary and sufficient coefficient conditions for functions to be in the class N .