Analytic structure in spaces of Lipschitz functions

Let $U\subseteq \mathbb{C}$ be bounded and open. For $0<\alpha <1$, $A_{\alpha }(U)$ is the set of functions in the little Lipschitz class with exponent $\alpha$ that are analytic in a neighborhood of $U$. We consider three conditions, motivated by the properties of bounded point derivations, that show how the functions in $A_{\alpha }(U)$ can have additional analytic structure than would otherwise be expected. We prove an implication between conditions $(c)$ and $(b)$ and show that there is no implication between conditions $(a)$ and $(c)$.