A sharp inequality for transport maps in via approximation

For f convex and increasing, we prove the inequality ∫ f ( | U ′ | ) ≥ ∫ f ( n T ′ ) , every time that U is a Sobolev function of one variable and T is the non-decreasing map defined on the same interval with the same image measure as U , and the function n ( x ) takes into account the number of pre-images of U at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.