Classes of univalent and multivalent Schwarz-Christoffel integrals and their applications
The authors present the scheme of proof of the following theorem: Given a set of reals \(\{\alpha_k\}^n_{k=1}\) satisfying the conditions \(0<\alpha_k<2\), \(\sum^n_{k=1} \alpha_k =n-2\), there exists an ordered set of distinct complex numbers \(\{a_k\}^n_{k=1}\), \(a_k =e^{i\varphi_k}\), \(0\leq \varphi_1< \varphi_2< \cdots <\varphi_n <2\pi\), such that the function \( f(z)= C_1\int^z_0 \prod^n_{k=1} (z-a_k)^{\alpha_k-1} dz+C_2\), \(C_1\neq 0\); \(C_1, C_2\in \mathbb{C}\) univalently maps the closed disk \(\overline E= \{z:| z|\leq 1\} =E\cup \partial E\) onto the polygon \(\overline D_n(\alpha_1, \alpha_2, \dots, \alpha_n)\). The result is applied to univalent solvability of the inverse boundary value problems with polygonal boundary, to the univalent variation of the Cisotti formula and classes of multivalent solutions of the Hilbert boundary value problem.
scientific article; zbMATH DE number 1175166