On functions that satisfy a weakened asymptotic monogeny condition
Given a complex function \(f\) in a domain \(D\), the following theorem is proved: if there exists a positive number \(\sigma\), such that \(f\) is monogenic at any \(\xi\in D\) relatively some set \(G(\xi)\) with the lower density at \(\xi\) greater than \(\sigma+\frac{1}{2}\) then one can find \(p_{\sigma\geq 1}\) such that local summability of the function \((\log^+|f(z)|)^{p_{\sigma}}\) implies that \(f\) is holomorphic in \(D\), with the asymptotic estimate \(p_{\sigma}=O(|\log\sigma|\sigma^{-\frac{3}{2}})\) for \(\sigma\to +0\) and for \(\sigma\geq\frac{11}{24}\) one can take \(p_{\sigma}:=1\). This paper continues the author's research on relations between asymptotic monogeneity and analyticity of complex functions.
scientific article; zbMATH DE number 1528693