Contractions and expansion

Let A⊆R be a finite set and let K⩾1 be a real number. Suppose that for each a∈A we are given an injective map ϕa:A→R which fixes a and contracts other points towards it in the sense that |a−ϕa(x)|⩽1K|a−x| for all x∈A, and such that ϕa(x) always lies between a and x. Then |⋃a∈Aϕa(A)|⩾K10|A|−OK(1). An immediate consequence of this is the estimate |A+K⋅A|⩾K10|A|−OK(1), which is a slightly weakened version of a result of Bukh.