A Generalization Of Yang Hongcang's Inequality

Laplacian operator is a linear operator.It is the most important operatorin differential geometry.Because many important nonlinear operators reduce t -o a Laplacian operator of some Riemannian metric after they were linearrization.Let D be a connected bounded domain in an n-dimentional Euclidean space Rⁿ.Assume that0<λ₁<λ₂≤λ₃≤L≤λ_k≤L,are eigenvalues of Laplacian operator with any order l for the Dirichlet problem:Where l∈N⁺,n is the unit outward normal to (?)D.Then we obtain an uppe -r bound of the (k + 1)-th eigenvalueλ_k+1 in terms of the first k eigenvalues.T -his inequality is independent of domain D ,that is, we prove the following:u_i is the corresponding eigenfunction with eigenvalueλ_i.When l is 1,we re -cover Yang Hongcang's inequality.