Radii problems for sections of convex functions

Abstract

A classical theorem of Szegö states that the sections f n ( z ) = z + ∑ k = 2 n a k z k {f_n}(z) = z + \sum \nolimits _{k = 2}^n {{a_k}{z^k}} of a convex function f ( z ) = z + ∑ k = 2 ∞ a k z k f(z) = z + \sum \nolimits _{k = 2}^\infty {{a_k}{z^k}} must be convex for | z | > 1 4 \left | z \right | > \frac {1}{4} . We determine disks | z | > r n \left | z \right | > {r_n} in which f n {f_n} is starlike and starlike of a positve order. Our proofs rely on some properties of convolutions.