On Bloch and automorphic functions (Q1378609)

Let \(D\) denote the unit disk and let \(\Gamma\) be a Fuchsian group of Möbius transformations acting on \(D\) and let \(F\) be a fundamental polygon of \(\Gamma\). A function \(f\) analytic on \(D\) is called automorphic with respect to \(\Gamma\) if \(f(\gamma(z))= f(z)\) for all \(\gamma \in \Gamma\), \(z\in D\) and additive automorphic with respect to \(\Gamma\) if \(f(\gamma(z))= f(z) + A_{\gamma}\) for all \(\gamma \in \Gamma\), \(z\in D\). The authors consider a variant of Bers' conjecture which states that if \(f\) is an analytic additive automorphic function with respect to \(\Gamma\) and if for \(p\geq 2\) the condition (1): \(\int\int_{F}(1-|z|^{2})^{p-2}|f'(z)|^{p} dx dy <\infty \), then \(f\) is a Bloch function. Recall a function \(f\) analytic in \(D\) is a Bloch function (written \(f\in B\)) if \(\sup_{z\in D} (1-|z|^{2})|f'(z)| <\infty \). Bers' conjecture was disproved by construction in the case \(p=2\) by Pommerenke. Here the authors consider instead the class \(f\) of automorphic functions with respect to \(\Gamma\) satisfying condition (1) and establish that \(f\) is a Bloch function. In fact, they prove two stronger results. First under condition (1) they show that \(f\) is actually a little Bloch function and secondly for the case \(p>2\) they weaken condition (1) and still prove \(f\in B\). Here an analytic automorphic function \(f\) with respect to \(\Gamma\) is called little Bloch if \((1-|z|^{2})|f'(z)| \to 0\) as \(z\to \partial D,\) and \(z\in F\). At the close of the paper the authors examine an analogue of Baernstein's result that for univalent functions \(f\) without zeros, one has \(\log f\in \text{BMOA}\). Here the authors prove for an analytic mean \(p\)-valent function \(f\) without zeros that \(\log f \in B\).