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Characterizations of sum form information measures on open domains

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Summary

The goal of this paper is to give a survey of all important characterizations of sum form information measures that depend uponk discrete complete probability distributions (without zero probabilities) of lengthn and which satisfy a generalized additivity property. It turns out that most of the problems have been solved, but some open problems lead to the very simple looking functional equations

$$f(pq) + f(p(1 - q)) + f((1 - p)q) - f((1 - p)(1 - q)) = 0, p,q \in ]0, 1[^k (FE)$$

and

$$f(pq) + f(p(1 - q)) + f((1 - p)q) - f((1 - p)(1 - q)) = g(p)g(q), p,q \in ]0, 1[^k , (LI)$$

wheref, g: ]0, , 1[k → ℝ andk ∈ ℕ. Moreover new entropies analogous to the Shannon entropy, entropies of degree α, entropies of degree (α, β) are introduced for χ, β ∈ ℕ.

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Ebanks, B.R., Kannappan, P., Sahoo, P.K. et al. Characterizations of sum form information measures on open domains. Aequ. Math. 54, 1–30 (1997). https://doi.org/10.1007/BF02755443

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