Aequationes Mathematicae 43 (1992) 219-224 0001-9054/92/030219-06 $1.50 + 0.20/0 University of Waterloo O 1992 Birkh/iuser Verlag, Basel Functional inequality characterizing nonnegative concave functions in (0, oo)l, JANUSZ MATKOWSKI Summary. In the present note we prove that every function f: (0, m)~ [0, m) satisfying the inequality af(s) + bf(t) <~f(as + bt), s, t > O, for some a and b such that 0 < a < 1 < a + b must be of the formf(t) =f(1)t, (t > 0). This improves our recent result in [2], where the inequality is assumed to hold for all s, t >I 0, and gives a positive answer to the question raised there. An analogue for functions of several real variables of the above result characterizes concave functions. Conjugate functions and some relations to H61der's and Minkowski's inequalities are mentioned. Introduction In a recent paper [2] we have proved that, without any regularity conditions, every function f: ~+ ~ I~+, (~+ ..= [0, oo)), satisfying the functional inequality af(s) + bf(t) <<.f(as + bt), (s, t >~0), for some a, b such that 0 < a < 1 < a + b must be of the form f(t) =f( l)t, (t >~ 0). It has also been shown that, using this result, one can get its analogue for functions of several real variables which, in turn, leads to a characterization of concave functions defined in Rk+, (k ~> 2), to a new concept of conjugate function and to a simultaneous generalization of H61der's and Minkowski's inequalities. The long proof of this result heavily depended upon the assumption that 0 belongs to the domain off. Nevertheless we conjectured that the theorem remains valid for every function f: (0, oo) ~ ~+. AMS (1980) subject classification (Revision 1985): Primary 39C05, 26B25, 26A51, Secondary 39B20. Manuscript received December 17, 1990, and in final form, August 9, 1991. 219