Radii problems for generalized sections of convex functions

Abstract:

A classical theorem of Szëgo states that for functions $f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}$ convex in $|z| < 1$, the sequence of partial sums ${f_n}(z) = z + \Sigma _{k = 2}^n{a_k}{z^k}$ must be convex in $|z| < \frac {1}{4}$. For the more general family consisting of functions of the form $z + \Sigma _{k = 2}^\infty {a_{{n_k}}}{z^{{n_k}}}$, where $\left \{ {{n_k}} \right \}$ denotes an increasing (finite or infinite) sequence of integers $( \geq 2)$, we find the radius of convexity $( \approx 0.21)$ and the radius of starlikeness $( \approx 0.37)$. The extremal function in both cases is $z + {z^2}/(1 - {z^2}) = z + \Sigma _{k = 1}^\infty {z^{2k}}$ associated with the convex function $z/(1 - z) = z + \Sigma _{k = 2}^\infty {z^k}$.