Uniformly locally univalent functions and Hardy spaces (Q1014661)

An analytic function \(f\) on the unit disk \(\mathbb D\) is called uniformly locally univalent if there exists \(\rho>0\) such that \(f\) is univalent on the hyperbolic disk \(|(z-a)/(1-\overline az)|<\tanh\rho\) for every \(a\in\mathbb D\). This is equivalent to \[ \|f''/f'\|=\sup_{z\in\mathbb D}(1-|z|^2)\left|\frac{f''(z)}{f'(z)}\right|<\infty. \] Let \(\mathcal A\) denote the set of analytic functions \(f\) on \(\mathbb D\), \(f(0)=0\), \(f'(0)=1\), and \[ \mathcal B(\lambda)=\{f\in\mathcal A: \|f''/f'\|\leq2\lambda\} \] for \(\lambda\geq0\). The authors compare the classes \(\mathcal B(\lambda)\) with the Hardy spaces \(H^p\). They prove Theorem 1: The inclusion \(\mathcal B(\lambda)\subset H^p\) does not necessarily hold for \(\lambda>1\) and \(0<p<1/(\lambda-1)\). Theorem 2: For \(\lambda\geq1\), \(\mathcal B(\lambda)\subset H^p\) for every number \(p\) with \(0<p<1/(\lambda^2-1)\). As a by-product of the proof, the authors state a relation between \(\mathcal B(\lambda)\) and spaces of Dirichlet type.