Universal inequalities for lower order eigenvalues of self-adjoint operators and the poly-Laplacian

In this paper, we first establish an abstract inequality for lower order eigenvalues of a self-adjoint operator on a Hilbert space which generalizes and extends the recent results of Cheng et al. (Calc. Var. Partial Differential Equations, 38, 409–416 (2010)). Then, making use of it, we obtain some universal inequalities for lower order eigenvalues of the biharmonic operator on manifolds admitting some special functions. Moreover, we derive a universal inequality for lower order eigenvalues of the poly-Laplacian with any order on the Euclidean space.