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April 2005 A Generalization of Bertilsson's Theorem
Oliver Roth, Karl-Joachim Wirths
Bull. Belg. Math. Soc. Simon Stevin 12(1): 53-63 (April 2005). DOI: 10.36045/bbms/1113318129


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 $S$.


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Oliver Roth. Karl-Joachim Wirths. "A Generalization of Bertilsson's Theorem." Bull. Belg. Math. Soc. Simon Stevin 12 (1) 53 - 63, April 2005.


Published: April 2005
First available in Project Euclid: 12 April 2005

zbMATH: 1078.30019
MathSciNet: MR2134856
Digital Object Identifier: 10.36045/bbms/1113318129

Primary: 30C75

Keywords: Brennan's conjecture , Löwner's method , univalent functions , variational methods

Rights: Copyright © 2005 The Belgian Mathematical Society

Vol.12 • No. 1 • April 2005
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