On multi-Jensen functions and Jensen difference
Bull. Korean Math. Soc. 2008 Vol. 45, No. 4, 729-737
Printed December 1, 2008
Krzysztof Ciepli\'nski
Pedagogical University
Abstract : In this paper we characterize multi-Jensen functions $f:V^{n}\!\!\rightarrow W$, where $n$ is a positive integer, $V, W$ are commutative groups and $V$ is uniquely divisible by $2$. Moreover, under the assumption that $f:\mathbb{R}\rightarrow \mathbb{R}$ is Borel measurable, we obtain representation of $f$ (respectively, $f,\, g,\, h:\mathbb{R}\rightarrow \mathbb{R}$) such that the Jensen difference \[2f\left(\frac{x+y}{2}\right)-f(x)-f(y)\] (respectively, the Pexider difference \[2f\left(\frac{x+y}{2}\right)-g(x)-h(y))\] takes values in a countable subgroup of $\mathbb{R}$.
Keywords : multi-Jensen function, multi-additive mapping, stability, Jensen difference, Pexider difference
MSC numbers : Primary 39B72, 39B82, 39A70, 39B22, 39B52
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