Covering numbers and schlicht functions (Q2192410)

The class \(\mathcal S\) of schlicht functions which consists of all univalent (holomorphic) functions \(f\) on the unit disk in the complex plane such that \(f(0)=0\) and \(f'(0)=1\) becomes a compact metric space being equipped with the metric \[ d(f,g)=\sum_{n\ge1}\lambda_n \min\Big\{ 1,\max_{|z|\le r_n}|f(z)-g(z)|\Big\}, \] where \(\lbrace\lambda_n \rbrace\) is a positive sequence with \(\sum_n \lambda_n<\infty\) and \(\lbrace r_n\rbrace\) is a sequence in \((0,1)\) tending to 1. The covering number \(N_{\mathcal S}(\delta)\), \(\delta > 0\) is the minimal number of closed balls of radius \(\delta\) covering \(\mathcal S\). Assuming additionally that \(\lambda_n\asymp\lambda^n\) and \(1-r_n\asymp n^{-\alpha}\) for some \(\lambda\in(0,1)\) and \(\alpha > 0\) it is proved that \(\log^2(1/\delta)\lesssim \log N_{\mathcal S}(\delta) \lesssim \log^{2+\alpha}(1/\delta)\), \(\delta\to0^+\).