Minimal superadditive extensions of superadditive functions

Abstract

Introduction* A real valued function / is said to be superadditive on an inverval I — [0, a] if it satisfies the inequality f(x + y) i> f{x) + f(y) whenever x, y and x + y are in I. Such functions have been studied in detail by E. Hille and R. Phillips [1] and R. A. Rosenbaum [2]. In this paper we show that any superadditive function/ on I has a minimal superadditive extension F to the non-negative real line E, and then proceed to show that F inherits much of its behavior from the behavior of / . We deal primarily with superadditive functions which are continuous and non-negative. A simple example of a superadditive function on [0, a] is furnished by a convex function / with /(0) ^ 0. Also, if / is convex and /(0) = 0, then it is easy to verify that its minimal superadditive extension F is given by