Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds μk,d^* obtained by Kröger (1999) for the k-th Neumann eigenvalue of a convex domain Ω⊂ Rᵈ, we prove the following inequalities: for any k∈ N there exists a constant $C(k,d) >0$ such that D_Ω² μₖ(Ω) ≤ μk,d^* - C(k,d) a₂(Ω)²/D_Ω² where D_Ω is the diameter of $Ω$ and a₂(Ω) is the second largest semiaxis of the John ellipsoid of $Ω$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.