We characterize the validity of the Hardy-type inequality ∫ s ∞ h (z) d z p, u, (0, t)q, w, (0, ∞) ≤ c h θ, v (0, ∞), where 0 < p < ∞, 0 < q ≤ ∞, 1 < θ ≤ ∞, u, w, and v are weight functions on (0, ∞). Some fairly new discretizing and antidiscretizing techniques of independent interest are used.