On a Subclass of Analytic Functions Satisfying a Differential Inequality

Let A denote the class of all functions f analytic in the unit disc D := {z is an element of C : |z| < 1} with the normalization f (0) = 0 = f'(0) - 1. For lambda is an element of (0, 1] and complex mu, a subclass of functions f in A satisfying |f'(z)(z/f(z))(2) - mu| < lambda on D is considered. The conditions on lambda and mu such that the functions in this subclass to be univalent are given. It is shown that the subclass is preserved under rotation, dilation and conjugation. A necessary and sufficient condition for analytic functions to be in this subclass is given. With this, the sharp second coefficient bound and the growth theorem are determined.