Let U be a bounded open subset of the complex plane and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A_{\alpha }(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the set of functions analytic on U that also belong to the little Lipschitz class with Lipschitz exponent <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>. It is shown that if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_{\alpha }(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> admits a bounded point derivation at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x \in \partial U$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>∂</mi> <mi>U</mi> </mrow> </math></EquationSource> </InlineEquation>, then there is an approximate Taylor Theorem for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A_{\alpha }(U)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> at x. This extends and generalizes known results concerning bounded point derivations.

Approximate Taylor Theorem for Analytic Lipschitz Functions

期刊论文
发表日期：

2026年5月8日

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Stephen Deterding

摘要

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