Quasi-additive functions

Summary Let 0 < 1 and letX, Y be real normed spaces. In this paper we consider the following functional inequality:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: X Y. Mainly continuous solutions are investigated. In the case whereY = R some necessary and some sufficient conditions for this inequality are given.Let 0 <1. The following functional inequality has been considered in [5]:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: R R. It appeared that the solutions of this inequality have properties very similar to those of additive functions (cf. [1], [2], [3]). The inequality under consideration seems to be interesting also because of its physical interpretation (cf. [5]). In this paper we shall consider this inequality in a more general case, wheref is defined on a real normed space and takes its values in another real normed space.The first part of the paper concerns the general case; in the second part we assume that the range off is inR.