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In this paper, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator of the Kohn Laplacian on a bounded domain in the Heisenberg group Hn . We establish two inequalities for eigenvalues of this problem. One of them implies an explicit estimate for the upper bound of the (k + 1) -th eigenvalue in terms of the first k eigenvalues. Moreover, as a special case, we give some universal inequalities and estimates for eigenvalues of the bi-Kohn Laplacian. Mathematics subject classification (2010): Primary 35P15, 53C17; Secondary 58C40.

Universal inequalities for eigenvalues of quadratic polynomial operator of the Kohn Laplacian