On the Boundary Value Problem for a GeneralizedCauchy–Riemann Equation in Lens and Lune

In this paper, we consider the Schwarz boundary value problem for a higher-order complex partial differential equation in a particular circular lens and two related complementary lunes. First, by applying the parqueting-reflection principle, we obtain a covering for the complex plane. Based on the points derived through this principle, we introduce the T-type and poly-Schwarz operators and study their properties. In particular, we discuss the boundary behavior of these operators. Finally, using the properties of the T-type and poly-Schwarz operators, we investigate the Schwarz problem for the inhomogeneous generalized Cauchy–Riemann equation in the lens domain and provide an explicit solution.