Refining the sharp upper bounds μ₊, ₃^* 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) μ₊, ₃^* - 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).
Bucur et al. (Tue,) studied this question.