Lower estimate for the integral means spectrum for \(p=-1\) (Q2781246)

For a bounded univalent function \(f\) in the unit disk, let NEWLINE\[NEWLINE \beta(p,f)=\limsup_{r\to 1} \frac{\log \int |f^\prime (re^{i\theta})|^p d\theta} {\log \frac{1}{1-r}}. NEWLINE\]NEWLINE S. Rhode [see \textit{C. Pommerenke}'s book ``Boundary behaviour of conformal maps'' (1992; Zbl 0762.30001)] has proved that there exists an \(f\) such that \(\beta (-1,f)\geq 0.109\). In the paper under review it is proved that there exists an \(f\) for which \(\beta (-1,f)\geq 0.127\). The construction is rather technical and uses an idea of Pommerenke that involves successive compositions of univalent functions.