A criterion for quasi-simple plane curves (Q1409794)

An oriented closed curve \(\gamma\) in the complex \(\mathbb{C}\) is called quasi-simple if it represents the positive oriented boundary of a simply connected domain \(D\subset\mathbb{C}\). In the paper, a non-standard criterion for closed curves to be quasi-simple is given. Roughly speaking, the main result asserts that a sufficient condition is a possibility to decompose the curve such that the arcs are quasi-simple and the vertexes are attainable from \(\infty\). In the case of piecewise smooth curve conditions on the arcs are formulated in terms of behavior of normal vector. These results are used to establish the univalence of the extremal polynomials in the maximal range problem for simply connected domains.