ON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS AND RADIUS PROPERTIES

Let S denote the class of normalized univalent functions f in the unit disk Delta. One of the problems addressed in this paper is that of the F-radius in G when F, G subset of S, namely the maximum value of r(0) such that r(-1) f(rz) is an element of G for all f is an element of F and 0 < r <= r(0). The investigations are concerned primarily with the classes U and P(2) consisting of univalent functions satisfying vertical bar f'(z)(z/f(z)(2) - 1 vertical bar <= 1 and vertical bar(z/f(z)''vertical bar <= 2, respectively, for all vertical bar z vertical bar < 1. Similar radius properties are also obtained for a geometrically motivated subclass S-p subset of S. Several new sufficient conditions for f to be in the class U are also presented.