On a certain functional equation and its application to the Schwarz problem

Abstract

The Schwarz problem for J-analytic functions in an ellipse is considered. In this case, the matrix J is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter l. This parameter is determined by the Jordan basis of the matrix J. An analysis of the functional equation was performed. It is shown that for l 2 [0, 1], the solution of the Schwarz problem with matrix J exists uniquely in the Hölder classes in an arbitrary ellipse