Computational Methods and Function Theory 1 (2001), No. 2, 521--533
Copyright Heldermann Verlag 2001
Oliver Roth
roth@mathematik.uni-wuerzburg.de, Mathematisches Institut, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany.
Karl-Joachim Wirths
kjwirths@tu-bs.de, Institut für Analysis, TU Braunschweig, Pockelsstr. 14, 38106 Braunschweig, Germany.
Let $\mathcal{S}$ denote the class of normalized schlicht functions in the unit disk. We consider for $f \in \mathcal{S}$ and $\lambda<0$ the Taylor coefficients $a_n(\lambda,f)$ of $(f(z)/z)^{\lambda}$ and prove that $|a_n(\lambda,f)|\le |a_n(\lambda,k)|$ for every $f \in \mathcal{S}$ and every $1 \leq n \leq -\lambda +1$, where $k(z)=z (1-z)^{-2}$ is the Koebe function. We also give~a necessary condition such that the Koebe function maximizes the functional $$ \sum_{k=1}^n \sigma_k |a_k(\lambda,f)|^2 $$ in the class $\mathcal{S}$ for given weights $\sigma_k \in \mathbb{R}$. These results supplement and complement previous results due to de Branges, Hayman and Hummel and others. Our proofs are based on the L\"owner differential equation combined with optimal control methods.
Keywords: Taylor coefficients, univalent functions, Löwner's method, optimization.
MSC 2000: Primary 30C75; Secondary 49K15.
[FullText-pdf (208 K)] [FullText-ps (268 K)]