Zygfryd Kominek
1990 Demonstratio Mathematica
Let (X,+) be a commutative, 2-divisible group. Than every approximately subadditive functional (i.e. f» X -*• 1R, R denotes the real line, such that f(x+y)< f(x)+f(y)+ £ for some £>0 and all i,;el), is minorized by a Jensen functional. Moreover, if fs X-"-Ris approximately convex (in the sense of Jensen) function, then there exists convex (in the sense of Jensen) function hi X -»R and a constant K suoh that |f-h|-\$K. In 1952 D.H. Hyers and S.M. Ulam ([2]) proved the stability of the inequality
more » ... (tx+(l-t)y)^tf(x) + (l-t)f(y), te[o,l], x,y6D, where D is a convex subset of In 1984 P.W. Cholewa ([1]) constructed the example showing that the stability property fails to hold for e-J-convex functions. More precisely, Cholewa constructed a function fi D -»1 satisfying the inequality f^E+a^ M + + 1 for all x , yeD , and such that there is no J-convex function uniformly close to f, i.e. for every J-convex function g: D R the inequality |f(x) -g(x)| < M, x e D, fails to hold for any constant M >0. In this example the set DcRwas, in particular, a commutative semigroup divisible by 2 and f was, additionally, subadditive. Our Theorem 2 shows -155