Univalent functions in the Möbius invariant \(Q_K\) space
For \(K:[0,\infty)\to [0,\infty)\), the space \(Q_K\) consists of those analytic functions \(f\) in the unit disc \(\mathbb{D}\) for which \[ \|f\|^2_{Q_K}:=\sup_{a\in\mathbb{D}}\int_{\mathbb{D}}|f'(z)|^2K(g(z,a))dA(z)<\infty. \] In the paper under review it is shown that, under a suitable regularity conditions for \(K\), a univalent function belongs to \(Q_K\) if and only if \[ \sup_{a\in\mathbb{D}}\int_0^1 M^2_\infty (r,f\circ \varphi_a-f(a)) K'(\log(1/r))dr < \infty, \] where \(\varphi_a(z)=(a-z)/(1-\overline{a}z)\) and \(M_\infty(r,f)=\max_{0\leq \theta \leq 2\pi}|f(re^{i\theta})|\). In addition, the authors show that, under the same regularity conditions for \(K\), the space \(Q_K\) contains all univalent Bloch functions if and only if \[ \int_0^1 \left(\log\left(\frac{1+r}{1-r}\right)\right)^2 K'(\log(1/r))dr<\infty. \]
scientific article; zbMATH DE number 5997783
scientific article