On a functional equation connected to sum form nonadditive information measures on an open domain. III. (English) Zbl 0609.39006
[Cf. the authors’ paper in C. R. Math. Acad. Sci., Soc. R. Can. 7, 45-50 (1985; Zbl 0568.39004).]
The object of this paper is to determine all solutions \(f,g_ j,h_ k:]0,1[\times]0,\infty [\to {\mathbb{R}}\), measurable in each variable, of \[ \sum^{m}_{j=1}\sum^{n}_{k=1}f(p_ jq_ k,u_ jv_ k)=\sum^{m}_{j=1}p_ j^{\alpha}u_ j\sum^{n}_{k=1}h_ k(q_ k,v_ k)+\sum^{n}_{k=1}q_ kv_ k\sum^{m}_{j=1}g_ j(p_ j,u_ j) \] for all \(u_ j>0\), \(v_ k>0\), \(p_ j>0\), \(q_ k>0\) \((j=1,...,m\); \(k=1,...,n)\) with \(\sum^{m}_{j=1}p_ j=\sum^{n}_{k=1}q_ k=1\) and for fixed but arbitrary \(m\geq 3\), \(n\geq 3\).
The object of this paper is to determine all solutions \(f,g_ j,h_ k:]0,1[\times]0,\infty [\to {\mathbb{R}}\), measurable in each variable, of \[ \sum^{m}_{j=1}\sum^{n}_{k=1}f(p_ jq_ k,u_ jv_ k)=\sum^{m}_{j=1}p_ j^{\alpha}u_ j\sum^{n}_{k=1}h_ k(q_ k,v_ k)+\sum^{n}_{k=1}q_ kv_ k\sum^{m}_{j=1}g_ j(p_ j,u_ j) \] for all \(u_ j>0\), \(v_ k>0\), \(p_ j>0\), \(q_ k>0\) \((j=1,...,m\); \(k=1,...,n)\) 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 |
