RADII PROBLEMS FOR GENERALIZED SECTIONS OF CONVEX-FUNCTIONS

A classical theorem of Szego states that for functions [GRAPHICS] convex in \z\ < 1, the sequence of partial sums [GRAPHICS] must be convex in \z\ < 1/4. For the more general family consisting of functions of the form [GRAPHICS] where {n(k)} denotes an increasing (finite or infinite) sequence of integers (greater-than-or-equal-to 2), we find the radius of convexity (almost-equal-to 0.21) and the radius of starlikeness (almost-equal-to 0.37). The extremal function in both cases is [GRAPHICS] associated with the convex function [GRAPHICS].