Sharp Hardy-type inequalities with Lamb's constant

Let Ω be an n-dimensional convex domain with finite inradius δ ₀=x∈ Ω \,\,δ, where δ = \,\,dist (x, ∂ Ω), and let $(p,q)$ be a pair of positive numbers. For functions vanishing at the boundary of the domain and any ν ∈ [0, p/q] we prove the following Hardy-type inequality ∫ Ω|∇ f|²δ ᵖ⁻¹ \,dx \,\,≥ \,\,h ∫Ω|f|²δ ᵖ⁺¹ \,dx + λ²/δ₀q ∫Ω|f|²δ p-q+1 \,dx with two sharp constants h=p²-ν² q²/4≥ 0 \,\,\,\,\,\,and \,\,\,\,\,\,λ = q/2 λν(2p/q) > 0, where z = λν(p) is the Lamb constant defined as the first positive root of the equation p Jν(z)+ 2 z Jν'(z)=0 for the Bessel function Jν. We prove that z = λν(p) as a function in p can be found as the solution of an initial value problem for the differential equation d z/dp=2z/p²-4ν² +4z². For $n=1$ our inequality is an improvement of the original Hardy inequality for finite intervals. For n≥ 1 and $p=q/2=1$ it gives a new sharp form of the Hardy-type inequality due to H. Brezis and M. Marcus. The case $h=0$, ν =1/2, $p=1$ and $q=2$ coincides with sharp eigenvalue estimates due to J. Hersch for $n=2$, and L. E. Payne and I. Stakgold for n≥ 3.