## 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
Full Text:

### References:

 [1] J. Aczél: Information functions on open domain III. C. R. Math. Rep. Acad. Sci. Canada 2 (1980), 281-285. [2] M. Behara, P. Nath: Additive and nonadditive entropy of finite measurable partitions. Probability and Information Theory, Vol. 2, Springer-Verlag 1973, Vol. 296, 102-138. · Zbl 0282.94020 [3] J. Havrda, F. Charvát: Quantification method of classification process, Concept of structural a-entropy. Kybernetika 3 (1967), 30-35. · Zbl 0178.22401 [4] Pl. Kannappan: On generalization of some measures in Information Theory. Glasnik Mat. 29 (1974), 81-93. · Zbl 0287.39006 [5] Pl. Kannappan: On some functional equations from additive and nonadditive measure - I. Proc. Edinburgh Math. Soc. 23 (1980), 145-150. · Zbl 0468.39002 [6] Pl. Kannappan, P. K. Sahoo: On a functional equation in two variables connected to sum form information measures on an open domain. · Zbl 0629.39006 [7] L. Losonczi: A characterization of entropies of degree a. Metrika 28 (1981), 237-244. · Zbl 0469.94005 [8] L. Losonczi, Gy. Maksa: On some functional equations of the information theory. Acta. Math. Acad. Sci. Hungar. 39 (1982), 73-82. · Zbl 0492.39006 [9] C. T. Ng: Information functions on open domains I, II. C. R. Math. Rep. Acad. Sci. Canada 2 (1980), 119-123 and 155-158. · Zbl 0446.94003 [10] P. K. Sahoo: On some functional equations connected to sum form information measures on open domains. Utilitas Math. 23 (1983), 161-175. · Zbl 0524.94008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.