Some results connected with Brennan's conjecture
Let \(f\) be a (univalent) conformal map of the unit disk \(\mathbb{D}\). For real \(t\) let \[ \beta_ f (t) = \limsup_{r \to 1} \log \int^{2 \pi}_ 0 \bigl | f' (re^{i \tau}) \bigr |^ t d \tau/ \log {1 \over 1 - r} \tag{*} \] and let \(B(t) = \sup_ f \beta_ f(t)\). \textit{J. E. Brennan} [J. Lond. Math. Soc., II. Ser. 18, 261-272 (1978; Zbl 0422.30006)] used a method of Carleson to prove that \(B(t) < | t |\) for \(t \leq - 2\) and he conjectured that \(B(t) = | t | - 1\). The present paper is a major step forward. The authors prove that there exists a universal constant \(t_ 0 \leq - 2\) such that \[ B(t) = | t | - 1 \quad \text{for} \quad - \infty < t \leq t_ 0. \tag{**} \] This is a great improvement of the best previously known result \(B(t) < | t | - 0.399\) for \(t \leq - 1\) [the reviewer, ibid. 32, 254-258 (1985; Zbl 0576.30017)]. The authors also show that \(B(t) > | t | - 1\) for \(-2 < t < - 1\); indeed they produce an example with a fractal boundary for which \(\beta_ f (t) > | t | - 1\) with \(\lim\sup\) replaced by \(\lim\inf\) in \((*)\). The proof of \((*)\) relies on an important refinement of \textit{L. Carleson}'s method [Duke Math. J. 40, 547-559 (1973; Zbl 0273.30014)]. They show that, for some absolute constant \(K (=2 | t_ 0 |\) above) and for all \(f\) with \(\infty \in f (\mathbb{D})\), there are at most const.\(\rho^{-K \varepsilon}\) disjoint disks \(B\) of radius \(\rho\) such that the harmonic measure of the \(B \cap \partial f (\mathbb{D})\) is \(> \rho^{1/2 + \varepsilon}\). The construction of the example uses a variational argument. The paper contains many further interesting results.
scientific article; zbMATH DE number 605116