Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions ai and bi, for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions ai and bi and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.