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Talk

Anderson's Conjecture and Radial Variation of Bloch Functions

  • Paul F.X. Müller (Linz)
A3 01 (Sophus-Lie room)

Abstract

In 1971 J.M. Anderson conjectured that for any conformal map $\varphi$ in the unit disc there exists $\beta , 0 \leq \beta \leq 2 \pi$ such that $$\int^{1}_{0} |\varphi`` (re^{i \beta }) | dr <\infty$$ More recently this problem has been posed in the works of N. Makarov, Ch. Pommerenke and D. Gnuschke - Ch. Pommerenke. The purpose of the present talk is to prove Anderson's conjecture. This will be done by showing the following theorem about the associated Bloch function $b=\log |\varphi`|$.
Theorem 1 There exists $\beta$,$0\leq \beta \leq 2 \pi$ such that $$b \left( re^{i \beta} \right) \leq -\delta \int^{r}_{0} |\nabla b \left( \rho e^{i \beta} \right) | d \rho + \frac{1}{\delta}$, for $0<r<1$$ where $\delta >0$ is independent of $r<1$.

seminar
26.11.96 30.01.25

Oberseminar Analysis

MPI für Mathematik in den Naturwissenschaften Leipzig (Leipzig) E2 10 (Leon-Lichtenstein) E1 05 (Leibniz-Saal)
Universität Leipzig (Leipzig) Augusteum - A314