Anderson's conjecture for domains with fractal boundary (Q5928679)

If \(f\) is a univalent function in the unit disk \(\mathbb D\), a conjecture of \textit{J. M. Anderson} [J. Reine Angew. Math. 249, 83-91 (1971; Zbl 0222.46036)] states that there exists some \(\zeta \in \mathbb T = \partial \mathbb D\) such that NEWLINE\[NEWLINE \int_0^1 |f''(r\zeta)|dr < \infty . \tag{1} NEWLINE\]NEWLINE Although this conjecture was recently proved by P. W. Jones and P. F. X. Müller, the problem of the size of the set on which \(f'\) has finite radial variation remains open. It is expected that the conjecture should hold for a set with Hausdorff dimension one. In this article it is shown that the set has the expected size, if the image domain \(f(\mathbb D)\) has fractal boundary. For that purpose, the author considers the space \(\mathcal B\) of Bloch functions, that is the space of holomorphic functions \(f\) in \(\mathbb D\) such that NEWLINE\[NEWLINE \|f\|_{\mathcal B} = \sup_{z \in \mathbb D}{(1-|z|^2)|f'(z)|} < \infty . NEWLINE\]NEWLINE It is well-known that if \(g\) is a univalent function in \(\mathbb D\) and \(f=\log{g'}\), then \(f \in \mathcal B\). Now, for positive numbers \(R\) and \(\varepsilon\) the author considers Bloch functions \(b\) with the property \(M(\varepsilon,R)\), that is NEWLINE\[NEWLINE \inf_{z \in \mathbb D}{\left(\sup_{w \in D_R(z)}{(1-|w|^2)|b'(w)|}\right)} > \varepsilon , NEWLINE\]NEWLINE where \(D_R(z) \subset \mathbb D\) denotes the disk with hyperbolic center \(z\) and hyperbolic radius \(R\). Then it is shown that if \(b \in \mathcal B\) has \(M(\varepsilon,R)\), then there exists a set \(E \subset \mathbb T\) of Hausdorff dimension one such that NEWLINE\[NEWLINE \liminf_{r \to 1}{\frac{\text{Re}{b(r\zeta)}} {\int_0^r |b'(\varrho\zeta)|d\varrho}} > 0 \tag{2} NEWLINE\]NEWLINE for all \(\zeta \in E\). Finally, if \(f\) is a univalent function in \(\mathbb D\) and \(f(\mathbb D)\) has fractal boundary then \(b=\log{f'} \in \mathcal B\) has \(M(\varepsilon,R)\) and (2) implies (1) for all \(\zeta \in E\).