Some functional inequalities |
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Authors: | Dr. Wolfgang Sander |
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Affiliation: | 1. Institut für Analysis, Technische Universit?t Braunschweig, Pockelsstrasse 14, D-3300, Braunschweig, Federal Republic of Germany
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Abstract: | LetM, N, O be open subsets of ? n and letF:M×N→O,f:O→?,g: M→?,h: N→? 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 ? 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. |
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