Some functional inequalities

Abstract

LetM, N, O be open subsets of ℝ n and letF:M×N→O,f:O→ℝ,g: M→ℝ,h: N→ℝ be functions, satisfying the functional inequality $$\forall (x,y) \in M \times N:f[F(x,y)] \leqslant g(x) + h(y).$$ IfF belongs to a certain extensive class of functions, we prove in this note, thatf is bounded above on every compact subset of ℝ n , wheneverh is bounded above on a Lebesgue-measurable set of positive Lebesgue-measure, contained inN (no assumptions aboutg are needed). Moreover we give applications of this theorem to generalized convex and subadditive functions.