Subadditive functions and partial converses of Minkowski ’ s and Mulholland ’ s inequalities

Let φ be an arbitrary bijection of R+. We prove that if the two-place function φ−1[φ(s) + φ(t)] is subadditive in R+ then φ must be a convex homeomorphism of R+. This is a partial converse of Mulholland’s inequality. Some new properties of subadditive bijections of R+ are also given. We apply the above results to obtain several converses of Minkowski’s inequality. Introduction. Throughout this paper R, R+, and N will stand respectively for the set of reals, nonnegative reals, and positive integers. Every function f : R+ → R satisfying the inequality f(s + t) ≤ f(s) + f(t) (s, t ≥ 0) is said to be subadditive. If the inequality is reversed the function is termed superadditive. In our recent paper [4] we have proved the following Theorem 1. If f : R+ → R+ is subadditive, right-continuous at 0 and bijective then f is a homeomorphism of R+. In Section 1 we consider the two-place function pφ : R+→R+ given by the formula pφ(s, t) := φ−1[φ(s) + φ(t)] where φ : R+→R+ is an arbitrary bijection. Using Theorem 1 we prove that if pφ is subadditive in R+ then φ is a convex homeomorphism of R+. This is a partial converse of Mulholland’s criterion of subadditivity of the functional pφ (cf. H. P. Mulholland 1991 Mathematics Subject Classification: Primary 26D15, 46E30, 39C05; Secondary 26A51, 26A18.