Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space R.Assume that 0<λ≤λ≤…≤λ≤…are the eigenvalues of the Dirichlet Laplacian operator with any order l: （-△）u=λu,in D u=■=…=■=0,on■■D. Then we obtain an upper bound of the（k+1）-th eigenvalueλin terms of the first k eigenvalues. sum from i=1 to k(λ-λ)≤（1/n）[4l（n+2l-2）]{sum from i=1 to k(λ-λ)λsum from i=1 to k(λ-λ)λ}. This ineguaiity is independent of the domain D.Furthermore,for any l≥3 the above inequality is better than all the known results.Our rusults are the natural generalization of inequalities corre- sponding to the case l=2 considered by Qing-Ming Cheng and Hong-Cang Yang.When l=1,our inequalities imply a weaker form of Yang inequalities.We aslo reprove an implication claimed by Cheng and Yang.