Note on the Ahlswede-Daykin inequality

Abstract

Given a finite boolean lattice L and four functions α, β, γ, δ of L to the nonnegative reals with α( x) β( gg)⩽ γ( x ∨ y) δ( x ∧ y) for all x, y ∈ L. We show that ∑ xϵX α(x)β(x′)⩽ ∑ rϵXvX′ γ(r)σ(r′) forx⊆L . Here x′ denotes the complement of x and X ∨ X′ stands for { x 1 ∨ x′ 2 | x 1, x 2 ∈ X}. This inequality turns out to be equivalent in a certain sense to the Ahlswede-Daykin inequality. We also apply our basic ideas to a special case of a conjecture of Fürstenberg.