Univalence of harmonic functions, problem of Ponnusamy and Sairam, and constructions of univalent polynomials

The criteria of the univalence of a harmonic mapping is obtained in this paper. Particularly, it permits to formulate the hypothesis of the harmonic function classes equality $S_H^0=S_H^0(S)$ (hypothesis of Ponnusamy and Sairam), in analytic form. The method of construction of the univalent harmonic polynomials with desired properties, according to a given harmonic function, is obtained by means of the univalence criteria.