Nehari type theorems and uniform local univalence of harmonic mappings

The paper is dealing with the criteria for uniform local univalence of the sense-preserving harmonic in the unit disc of complex plane functions in terms of generalised Schwarzian derivative introduced by R. Hernández and M. J. Martín. The main section is devoted to the proof of conditions of univalence and uniform local univalence by the means of estimation of generalized Schwarzian derivatives and methods of theory of linear-invariant families. The proved criteria are effective in the case of quasiconformal harmonic functions that was confirmed by examples. In the final section some related methods are applied to the harmonic functions associated with the minimal graphs. The estimation of Gaussian curvature of minimal surfases is obtained in the terms of order of associated harmonic function.