Integrability of the derivative of the Riemann mapping function for wedge domains (Q1101573)

Let \(\Gamma\) be a closed Jordan curve in the complex plane with interior \(\Omega\) and exterior \(\Omega^*\). Let \(\alpha\in (0,1)\) and \(r>0\). \(\Gamma\) is said to satisfy the interior \(\alpha\)-r wedge condition if for very \(w\in \Gamma\), a closed circular sector of opening \(\alpha\pi\) and radius r lies in \({\bar \Omega}\), with vertex at w. If the same is true with \(\Omega^*\) substituted for \(\Omega\), \(\Gamma\) is said to satisfy the exterior \(\alpha\)-r wedge condition. Let f be a conformal mapping of the unit disc onto \(\Omega\). Theorem 1. Suppose that \(\Gamma\) satisfies an exterior \(\alpha\)-r wedge condition. Then \(1/f'\in H^ p\) for all \(p<1/2(1-\alpha)\). Theorem 2. Suppose that \(\Gamma\) satisfies an interior \(\alpha\)-r wedge condition. Then \(f'\in H^ p\) for all \(p<1+\alpha /2(1-\alpha)\).