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On a functional equation in two variables connected to sum form information measures on an open domain. (English) Zbl 0629.39006

The equation (1) \(\sum^{n}_{i=1}\sum^{m}_{j=1}f_{ij}(p_ iq_ j,r_ is_ j)=\sum^{n}_{i=1}g_ i(p_ i,r_ i)+\sum^{m}_{j=1}h_ j(q_ j,s_ j)\) for \(f_{ij}=g_ i=h_ j=f\) arises in connection with the axiomatic characterization of directed divergence and inaccuracy in information theory. We determine the measurable solutions of the functional equation (1) for an arbitrary but fixed pair of integers \(n\geq 2\), \(m\geq 3\) on an open domain.

MSC:

39B99 Functional equations and inequalities
94A17 Measures of information, entropy
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