A functional inequality characterizing convex functions, conjugacy and a generalization of Hölder's and Minkowski's inequalities

Summary

The main result says that, iff: ℝ₊ → ℝ₊ satisfies the functional inequalityaf(s) + bf(t) ⩽ f (as + bt) (s,t ⩾ 0) for somea, b such that 0 <a < 1 <a + b, thenf(t) = f(1)t, (t ⩾ 0). A relevant result for the reverse inequality is also discussed. Applying these results we determine the form of all functionsf: ℝ ⁺_k → ℝ₊ satisying the above inequalities. This leads to a characterization of both concave and convex functions defined on ℝ ^k−1₊ , to a notion of “conjugate functions” and to a general inequality which contains Hölder's and Minkowski's inequalities as very special cases.