For 1 ⩽ p ⩽ ∞ 1 \leqslant p \leqslant \infty , we consider p p -integrable functions on a finite cube Q 0 {Q_0} in R n {{\mathbf {R}}^n} , satisfying \[ ( 1 | Q | ∫ Q | f ( x ) − f Q | p d x ) 1 / p ⩽ C φ ( | Q | ) {\left ( {\frac {1} {{|Q|}}\int _Q {|f(x) - {f_Q}{|^p}dx} } \right )^{1/p}} \leqslant C\varphi (|Q|) \] for every parallel subcube Q Q of Q 0 {Q_0} , where | Q | |Q| denotes the volume of Q Q , f Q {f_Q} is the mean value of f f over Q Q and φ ( t ) \varphi (t) is a nonnegative function defined in ( 0 , ∞ ) (0,\infty ) , such that φ ( t ) \varphi (t) is nonincreasing near zero, φ ( t ) → ∞ \varphi (t) \to \infty as t → 0 t \to 0 , and t φ p ( t ) t{\varphi ^p}(t) is nondecreasing near zero. The constant C C does not depend on Q Q . Let g g be a nonnegative p p -integrable function g : ( 0 , 1 ) → R g:(0,1) \to {\mathbf {R}} such that g g is nonincreasing and g ( t ) → ∞ g(t) \to \infty as t → 0 t \to 0 . We prove here that there exist a cube Q 0 {Q_0} and a function f f satisfying condition ( 1 ) (1) for every parallel subcube Q Q of Q 0 {Q_0} , such that δ f ( λ ) ⩾ C 1 δ g ( λ ) {\delta _f}(\lambda ) \geqslant {C_1}{\delta _g}(\lambda ) for λ ⩾ λ 0 \lambda \geqslant {\lambda _0} , C 1 > 0 {C_1} > 0 , where δ ( λ ) \delta (\lambda ) denotes the distribution function.