Radii problems for generalized sections of convex functions

Abstract

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