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 $\partial _{{\overline{z} }} - {\upbeta} \frac{Z}{{\overline{Z} }}\partial _{z} ,\;0 \leq {\upbeta} < 1$).

It is proved that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral

$${\int\limits_{{\upgamma}\backslash {\left\{ {{\upzeta} \in {\upgamma} :{\left| {{\upzeta} - t} \right|} \leq r} \right\}}} {\frac{{{\left| {f{\left( {\upzeta} \right)} - f{\left( t \right)}} \right|}}}{{{\left| {{\upzeta} - t{\left| {t \mathord{\left/ {\vphantom {t {\upzeta}}} \right. \kern-\nulldelimiterspace} {\upzeta}} \right|}^{{\uptheta}} } \right|}}}{\left| {{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta}}{{\overline{{\upzeta}} }}\overline{n} {\left( {\upzeta} \right)}} \right)}} \right|}ds,\quad {\uptheta} = \frac{{2{\upbeta}}}{{1 - {\upbeta}}},} }$$

converges uniformly on γ as r → 0, where n(ζ) is the unit vector of outer normal on γ at a point ζ and ds is the differential of arc length, then the β-Cauchy-type integral

$$\frac{1}{{2{\left( {1 - {\upbeta} } \right)}{\uppi} }}{\int\limits_{\mathrm \gamma} {\frac{{f{\left( {\upzeta} \right)}}}{{{\upzeta} - z{\left| {z \mathord{\left/ {\vphantom {z {\upzeta} }} \right. \kern-\nulldelimiterspace} {\upzeta} } \right|}^{\uptheta } }}{\left( {n{\left( {\upzeta} \right)} - {\upbeta} \frac{{\upzeta} }{{\overline{{\upzeta} } }}\overline{n} {\left( {\upzeta} \right)}} \right)}ds,\quad z \notin {\upgamma} ,} }$$

admits a continuous extension to γ and a version of the Sokhotski–Plemelj formulas holds.