Univalence Constraints on the Schwarz–Christoffel Parameters

The Area Theorem and coefficient inequalities for univalent functions are used to derive inequality constraints on the accessory parameters in the Schwarz–Christoffel formula for the conformal mapping of the unit disk onto the interior or onto the exterior of a polygon. The constraints restrict the choice of prevertices and are necessary for the univalence of the mapping. The conditions are explicit, easy to compute and are applicable to the numerical computation of the Schwarz–Christoffel map. The relative effectiveness of the constraints is illustrated by elementary examples.