Radii problems for sections of convex functions

Abstract:

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