Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian (-Delta)(s)|Omega for s > 0 and s is an element of Q(+). Here Omega subset of R-n is a bounded open domain, and Q+ is the set of all positive rational numbers. Secondly, if s. Q+ and s >= 1 (in this case, the operator is also called the non-integer poly-Laplacian), then by this universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called Yang type inequality for the corresponding eigenvalue problem, which is the extension to the case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding eigenvalues from the Yang type inequality.