The U-Radius for Classes of Analytic Functions

Let U denote the class of normalized analytic functions f in the open unit disk D satisfying vertical bar(Z/f(Z))(2) f' (Z) - 1 vertical bar < 1. The U-radius is obtained for several classes of functions. These include the class of normalized analytic functions f satisfying the inequality Re f (z)/g(z) > 0 or vertical bar f (z)/g(z) -1 vertical bar < 1 in D, where g belongs to a certain class of functions, the class of functions f satisfying vertical bar f' (z) - 1 vertical bar < 1 in D, and functions f satisfying Re f (z)/z > alpha, 0 <= alpha < 1, in D. A recent conjecture by Obradovic and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true.