We are concerned with the following problem. Let $L$
and $K$ be fixed real numbers. When does the Koebe function $k(z)=z (1-z)^{-2}$
maximize the $N$th Taylor coefficient of $(1/f'(z))^L(z/f(z))^K$
for $f$ in the class $S$ of normalized schlicht functions? A sufficient
condition for $L \geq -1$ is $1 \leq N
\leq 2L+K+1$. A necessary condition is that a certain trigonometric sum
involving hypergeometric functions is non-negative. These results generalize a
recent theorem of Bertilsson and suggest a link between Brennan's conjecture in
conformal mapping and Baernstein's theorem about integral means of functions in
$S$.
@article{1113318129,
author = {Roth, Oliver and Wirths, Karl-Joachim},
title = {A Generalization of Bertilsson's Theorem},
journal = {Bull. Belg. Math. Soc. Simon Stevin},
volume = {11},
number = {5},
year = {2005},
pages = { 53-63},
language = {en},
url = {http://dml.mathdoc.fr/item/1113318129}
}
Roth, Oliver; Wirths, Karl-Joachim. A Generalization of Bertilsson's Theorem. Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2005) no. 5, pp. 53-63. http://gdmltest.u-ga.fr/item/1113318129/