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In this paper, we prove some analogues of Payne-Polya-Weinberger,Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplaceeigenvalues on any subset in the integer lattice $\Z^n.$ This partially answersa question posed by Chung and Oden.

Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet Laplace eigenvalues on integer lattices