Unified Poincaré and Hardy inequalities with sharp constants for convex domains

Let 0 be an n-dimensional convex domain, and let v ∈ [0,1/2]. For all f ∈ H 1 0 (Ω) we prove the inequality ∫|∇f| 2 dx ≥ (1 4 - v 2 ) ∫ Ω |f| 2 δ 2 dx + λ 2 v δ 2 0 ∫ Ω |f| 2 dx, where δ = dist(x, ∂Ω), δ 0 = sup δ. The factor λ 2 v , is sharp for all dimensions, λ v being the first positive root of the Lamb type equation J v (λ v ) + 2λ v J' v (λ v ) = 0 for Bessel's functions. In particular, the case v = 0 with λ 0 = 0,940... presents a new sharp form of the Hardy type inequality due to Brezis and Marcus, while in the case v = 1/2 with λ 1/2 = π/2 we obtain a unified proof of an isoperimetric inequality due to Poincare for n = 1, Hersch for n = 2 and Payne and Stakgold for n > 3. A generalization, when the latter integral is replaced by the integral ∫ Ω |f| 2 / δ 2-m dx, m > 0, is proved, too. As a special case, we obtain the sharp inequality ∫ Ω |f| 2 dx ≥ m 2 j 2 1 /m-1 4δ m 0 ∫ Ω |f| 2 δ 2-m dx, where j v is the first positive zero of J v .