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,R_ρ (z) : = under(∑, k) (z - λ_k)₊^ρ . Here {λ_k}_{k = 1}^∞ are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ R^d, and x₊ : = max (0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λ_k, on averages such as over(λ_k, -) : = frac(1, k) ∑_{ℓ ≤ k} λ_ℓ, and on the eigenvalue counting function. For example, we prove that for all domains and all k ≥ j frac(1 + frac(d, 2), 1 + frac(d, 4)),frac(over(λ_k, -), over(λ_j, -)) ≤ 2 (frac(1 + frac(d, 4), 1 + frac(d, 2)))^{1 + frac(2, d)} (frac(k, j))^{frac(2, d)} .