Approximate Taylor theorem for analytic Lipschitz functions

Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$ admits a bounded point derivation at $x \in \partial U$, then there is an approximate Taylor Theorem for $A_{\alpha}(U)$ at $x$. This extends and generalizes known results concerning bounded point derivations.