A property of the β-Cauchy-type integral with continuous density

A theorem from the classical complex analysis proved by Davydov in 1949 is extended to the theory of solution of a special case of the Beltrami equation in the z-complex plane (i.e., null solutions of the differential operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _{{\overline{z} }} - {\upbeta} \frac{Z}{{\overline{Z} }}\partial _{z} ,\;0 \leq {\upbeta} < 1$$\end{document}).