Schwartz-Christoffel integrals reducibility to close-to-convex

Abstract

The possibility to find close-to-convex functions in a class of unit disk mapping onto bounded polygonal domain with rectilinear boundary by means of Schwartz-Christoffel integrals is investigated in the conditions of all possible exponents permutations. The notions of reducible and irreducible exponents collections and uniformly reducible set of collection are introduced. Linear programming problems series solving is offered for investigation of uniformly reducibility of set. An algorithm optimizing the computations by means of special subset of series construction is elaborated. The search of reducing permutation is replaced by search of special Hamiltonian cycle in complete graph. The sufficient conditions of reducibility and irreducibility are obtained. The examples of irreducible collections are constructed. The methods of constructions of new collections from old ones are given. For the above mentioned class of Schwartz-Christoffel integrals the Paatero's theorem about sufficient condition of univalence is generalized.