A generalization of Bertilsson’s theorem. (English) Zbl 1078.30019

Let \(S\) denote the class of functions \(f\) of the form \(f(z)= z + a_2 z^2+\cdots\), holomorphic and univalent in the disc \(D= \{z\in\mathbb{C}:|z|< 1\}\), and let \(k(z)= z(1- z)^{-2}\), \(z\in D\) (the Koebe function). The authors consider the following problem. When does the Koebe function maximize the \(N\)th Taylor coefficient of \((1/f'(z))^L(z/f(z))^K\) for \(f\in S\) where \(L\) and \(K\) are fixed real numbers? A sufficient condition for \(L\geq -1\) is \(1\leq N\leq 2L+ K+1\) (Theorem 1).
A necessary condition is that a certain trigonometric sum involving hypergeometric functions is non-negative (Theorem 2). These results generalize the known theorem of Bertilsson and suggest a link between the known Brennan’s conjecture in the class \(S\) and the Baernstein’s theorem about integral means of \(f\in S\). The proof of Theorem 1 is based on the Loewner differential equation and the proof of Theorem 2 – on an elementary case of Schiffer’s method of boundary variation.


30C75 Extremal problems for conformal and quasiconformal mappings, other methods
Full Text: Euclid