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 $z$, $w \in U$, $|f(z)-f(w)| \leq \epsilon |z-w|^{\alpha}$ 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.