Approximate Taylor theorem for analytic Lipschitz functions

TL;DR

This paper establishes an approximate Taylor theorem for functions in the Lipschitz class that are analytic on a domain, extending previous results related to bounded point derivations in complex analysis.

Contribution

It generalizes known results by proving an approximate Taylor theorem for analytic Lipschitz functions with bounded point derivations at boundary points.

Findings

Existence of approximate Taylor expansion under bounded point derivation conditions

Extension of classical results in complex analysis and Lipschitz function theory

Broader applicability to analytic functions with Lipschitz regularity

Abstract

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.