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Abstract. In this article, we determine the radius of univalence of sections of normalized univalent harmonic mappings for which the range is convex (resp. starlike, close-to-convex, convex in one direction). Our result on the radius of univalence of section s n,n (f ) is sharp especially when the corresponding mappings have convex range. In this case, each section s n,n (f ) is univalent in the disk of radius 1/4 for all n ≥ 2, which may be compared with classical result of Szegö on conformal mappings.

Abstract. The criterion of the univalence of a harmonic mapping is obtained in this paper. Particularly, it permits to formulate the conjecture of coincidence of the harmonic function classes S 0 H = S 0 H (S) (the problem 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.