Non-tangential limits for analytic Lipschitz functions

Abstract

Let $U$ be a bounded open subset of the complex plane. Let $0<α<1$ and let $A_α(U)$ denote the space of functions that satisfy a Lipschitz condition with exponent $α$ on the complex plane, are analytic on $U$ and are such that for each $ε>0$, there exists $δ>0$ such that for all $z$, $w \in U$, $|f(z)-f(w)| \leq ε|z-w|^α$ whenever $|z-w| < δ$. We show that if a boundary point $x_0$ for $U$ admits a bounded point derivation for $A_α(U)$ 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.