Fundamental equation of information revisited.

*(English)*Zbl 0842.39008Let \(J\) be the \(k\)-th Cartesian power of the open interval \(]0,1[\). The fundamental equation of information measures of multiplicative type, depending upon \(k\) probability distributions with nonzero probabilities, is \(f(x)+M(1-x)f(y/(1-x))=f(y)+M(1-y)f(x/(1-y))\) (whenever \(x\), \(y\) and also \(x+y\) are in \(J\)), where \(M: J\to \mathbb{R}\) is multiplicative. This equation has been completely solved by C. T. Ng and the reviewer [Linear Algebra Appl. 52-53, 1-30 (1983; Zbl 0517.39006)].

The author gives here an alternative proof for the case where \(M\) is also additive (as is the case for the most applied information measures) which uses a result of B. Jessen, J. Karpf and A. Thorup [Math. Scand. 22, 257-265 (1968; Zbl 0183.04004)] just once (instead of twice) and is more similar to the proof for the nonadditive case in the 1983 paper. Finally, generalizations of (part of) the result to more general domains (ordered commutative rings with unit, in particular real-closed fields and positive cones in ordered fields) are offered.

The author gives here an alternative proof for the case where \(M\) is also additive (as is the case for the most applied information measures) which uses a result of B. Jessen, J. Karpf and A. Thorup [Math. Scand. 22, 257-265 (1968; Zbl 0183.04004)] just once (instead of twice) and is more similar to the proof for the nonadditive case in the 1983 paper. Finally, generalizations of (part of) the result to more general domains (ordered commutative rings with unit, in particular real-closed fields and positive cones in ordered fields) are offered.

Reviewer: J.Aczél (Waterloo / Ontario)

##### MSC:

39B22 | Functional equations for real functions |

94A17 | Measures of information, entropy |

39B52 | Functional equations for functions with more general domains and/or ranges |

##### Keywords:

ordered commutative rings; fundamental equation of information measures; positive cones in ordered fields
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\textit{B. R. Ebanks}, Aequationes Math. 51, No. 1--2, 86--99 (1996; Zbl 0842.39008)

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##### References:

[1] | Aczél, J. andDaróczy, Z.,Measures of information and their characterizations. Academic Press, New York, 1975. · Zbl 0345.94022 |

[2] | Aczél, J. andNg, C. T.,Determination of all semisymmetric recursive information measures of multiplicative type on n positive discrete probability distributions. Linear Alg. Appl.52 (1983), 1–30. · Zbl 0517.39006 |

[3] | Jacobson, N.,Lectures on abstract algebra, vol. III. D. van Nostrand, Princeton, 1964. · Zbl 0124.27002 |

[4] | Jessen, B., Karpf, J. andThorup, A.,Some functional equations on groups and rings. Math. Scand.22 (1968), 257–265. · Zbl 0183.04004 |

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[6] | Lang, S.,Algebra. Addison-Wesley, Reading, MA, 1965. |

[7] | Maksa, Gy.,Solution on the open triangle of the generalized fundamental equation of information with four unknown functions. Utilitas Math.21 (1982), 267–282. · Zbl 0497.94003 |

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[9] | Rédei, L.,Algebra, vol. I. Pergamon Press, Oxford, 1967. |

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