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D E M O N S T R A T E MATHEMATICAVol. X X X I XNo 42006Michal LewickiA R E M A R K ON QUASIAFFINE FUNCTIONSA b s t r a c t . We consider the functional inequalitymin{/(x), f(y)} < f (rx + (1 - r)y) < max{/(x),(*)f{y)},where / is a real valued function on a linear space X and r £ (0,1) is fixed. The purposeof the present paper is to investigate connections between functions satisfying inequality(*) and solutions of (*) with r =As a conclusions we get, that under some regularityassumptions, function / is of the form / = g o a, where a : X —> R is an additive andg : R —• R is monotone.Let X be a linear space over R, D C X be a convex set and r G (0,1) befixed. A function / : D —» M is said to be r-affine if it satisfies(1)f(rx + (1 - r)y) = rf(x) + (1 - r)f(y),x,y G D.Clearly, if / is r-affine with some r € (0,1) then(2)min{/(aO, f(y)} < f(rx + (1 - r)y) < m a x { f ( x ) , f(y)},x,y e D.Functions
Demonstratio Mathematica – de Gruyter
Published: Oct 1, 2006
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