Favardʹs inequality on average values of convex functions

Let f ≥ 0 be a continuous, concave function on [a, b]. Let ƒ = (1/(b - a)) δab f(t)dt. Favardʹs inequality is that, when δ = f̄, 12δ∫ƒ−δƒ+δψ(u)du ≥ 1b − a∫ab ψ (ƒ(t)) dt for all convex functions, ψ, defined on (f̄ - δ, f̄ + δ). We show there is a δ for which inequality (1) is valid for a class of nonconvex functions ψ. Further, there is an optimal δ for which the reverse inequality of line (1) is true. The reverse inequality is strictly sharper (in this setting) then Jensenʹs inequality.