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On a functional equation connected to sum form nonadditive information measures on an open domain. I. (English) Zbl 0609.39005

The objective of this paper is to determine all measurable solutions f:]0,1[\(\to {\mathbb{R}}\) of \[ \sum^{m}_{j=1}\sum^{n}_{k=1}f(p_ jq_ k)=\sum^{m}_{j=1}f(p_ j)+\sum^{n}_{k=1}f(q_ k)+\lambda \sum^{m}_{j=1}f(p_ j)\sum^{n}_{k=1}f(q_ k) \] for all \(p_ j>0\) \((j=1,...,m)\), \(q_ k>0\) \((k=1,...,n)\) with \(\sum^{m}_{j=1}p_ j=\sum^{n}_{k=1}q_ k=1\) and for fixed (but arbitrary) \(m\geq 3\), \(n\geq 3\).
Reviewer: J.Aczél

MSC:

39B99 Functional equations and inequalities
39B62 Functional inequalities, including subadditivity, convexity, etc.
94A17 Measures of information, entropy
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References:

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