Quantitative Kröger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds $μ_{k,d}^* $ obtained by Kröger (1999) for the $k$-th Neumann eigenvalue of a convex domain $Ω\subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_Ω^2 μ_k(Ω) \leq μ_{k,d}^* - C(k,d) a