Sharp Hardy’s inequalities in Hilbert spaces

We study the behavior of the smallest possible constants d(a,b) and dₙ in Hardy’s inequalities ∫ₐᵇ(1/x∫ₐˣf(t)dt)²\:dx≤ d(a,b){}∫ₐᵇ [f(x)]²\: dx and ∑ₖ₌₁ⁿ(1/k∑ⱼ₌₁ᵏaⱼ)²≤ dₙ{}∑ₖ₌₁ⁿaₖ². The exact constant d(a,b) and the precise rate of convergence of dₙ are established and the extremal function and the “almost extremal” sequence are found.