Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,Rρ(z):=∑k(z−λk)+ρ. Here {λk}k=1∞ are the ordered eigenvalues of the Laplacian on a bounded domain Ω⊂RdΩ⊂Rd, and x+:=max(0,x)x+:=max(0,x) denotes the positive part of the quantity x  . As corollaries of these inequalities, we derive Weyl-type bounds on λkλk, on averages such as λk¯:=1k∑ℓ⩽kλℓ, and on the eigenvalue counting function. For example, we prove that for all domains and all k⩾j1+d21+d4,λk¯λj¯⩽2(1+d41+d2)1+2d(kj)2d.