Non-tangential limits for analytic Lipschitz functions

Abstract

Let U be a bounded open subset of the complex plane. Let $$0<\alpha <1$$ and let $$A_{\alpha }(U)$$ denote the space of functions that satisfy a Lipschitz condition with exponent $$\alpha $$ on the complex plane, are analytic on U and are such that for each $$\epsilon >0$$ , there exists $$\delta >0$$ such that for all z, $$w \in U$$ , $$|f(z)-f(w)| \le \epsilon |z-w|^{\alpha }$$ whenever $$|z-w| < \delta $$ . We show that if a boundary point $$x_0$$ for U admits a bounded point derivation for $$A_{\alpha }(U)$$ and U has an interior cone at $$x_0$$ then one can evaluate the bounded point derivation by taking a limit of a difference quotient over a non-tangential ray to $$x_0$$ . Notably our proofs are constructive in the sense that they make explicit use of the Cauchy integral formula.