G. Khuskivadze, V. Paatashvili

On a Representation of the Derivative of a Conformal Mapping

abstract:
Let $\om$ conformally map the unit circle on a plane singly-connected domain $D$ bounded by a simple rectifiable curve. It is shown that for the function $\lg\om'$ to be represented in the unit circle by a Cauchy type $A$-integral with density $\arg\om'$, it is necessary and sufficient that $D$ be a Smirnov domain. In particular, for this representation to be done by a Cauchy--Lebesgue type integral with the same density, it is necessary and sufficient that the function $\lg\om'$ belong to the Hardy class $H_1$