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A remark on quasiaffine functions

Lewicki M.

Demonstratio Mathematica

2006

Vol. 39, nr 4

743-750

We consider the functional inequality (*) min{f(x), f(y)} is less than or equal to f (rx + (1 - r)y) is less than or equal to max{f(x), f(y)}, where f is a real valued function on a linear space X and r is an element of (0, 1) is fixed. The purpose of the present paper is to investigate connections between functions satisfying inequality (*) and solutions of (*) with r = 1/2. As a conclusions we get, that under some regularity assumptions, function f is of the form f = g o alpha, where alpha : X approaches R is an additive and g : R approaches R is monotone.