K. Baron and Z. Kominek [2] have studied the functional inequality f(x+y) - f(x) - f(y) ≥ ϕ (x,y), x, y ∈ X, under the assumptions that X is a real linear space, ϕ is homogeneous with respect to the second variable and f satisfies certain regularity conditions. In particular, they have shown that ϕ is bilinear and symmetric and f has a representation of the form f(x) = ½ ϕ(x,x) + L(x) for x ∈ X, where L is a linear function. The purpose of the present paper is to consider this functional inequality under different assumptions upon X, f and ϕ. In particular we will give conditions which force biadditivity and symmetry of ϕ and the representation f(x) = ½ ϕ(x,x) - A(x) for x ∈ X, where A is a subadditive function.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-6,
author = {W\l odzimierz Fechner},
title = {On Functions with the Cauchy Difference Bounded by a Functional},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {52},
year = {2004},
pages = {265-271},
zbl = {1099.39018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-6}
}
Włodzimierz Fechner. On Functions with the Cauchy Difference Bounded by a Functional. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) pp. 265-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-3-6/