Bulletin of the Belgian Mathematical Society - Simon Stevin

A Generalization of Bertilsson's Theorem

Oliver Roth and Karl-Joachim Wirths

Full-text: Open access

Abstract

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 information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 12, Number 1 (2005), 53-63.

Dates
First available in Project Euclid: 12 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1113318129

Digital Object Identifier
doi:10.36045/bbms/1113318129

Mathematical Reviews number (MathSciNet)
MR2134856

Zentralblatt MATH identifier
1078.30019

Subjects
Primary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods

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

Citation

Roth, Oliver; Wirths, Karl-Joachim. A Generalization of Bertilsson's Theorem. Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 1, 53--63. doi:10.36045/bbms/1113318129. https://projecteuclid.org/euclid.bbms/1113318129


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