Let Ω be a fixed open cube in ℝ n . For r ∈[1, ∞) and α∈[0, ∞) we define formula here where Q is a cube in ℝ n (with sides parallel to the coordinate axes) and χ Q stands for the characteristic function of the cube Q . A well-known result of Gehring [ 5 ] states that if formula here for some p ∈(1, ∞) and c ∈(0, ∞), then there exist q ∈( p , ∞) and C = C ( p , q , n , c )∈(0, ∞) such that formula here for all cubes Q ⊂Ω, where [mid ] Q [mid ] denotes the n -dimensional Lebesgue measure of Q . In particular, a function f ∈ L 1 (Ω) satisfying (1.1) belongs to L q (Ω). In [ 9 ] it was shown that Gehring's result is a particular case of a more general principle from the real method of interpolation. Roughly speaking, this principle states that if a certain reversed inequality between K -functionals holds at one point of an interpolation scale, then it holds at other nearby points of this scale. Using an extension of Holmstedt's reiteration formulae of [ 4 ] and results of [ 8 ] on weighted inequalities for monotone functions, we prove here two variants of this principle involving extrapolation spaces of an ordered pair of (quasi-) Banach spaces. As an application we prove the following Gehring-type lemmas.