We present an axiomatic characterization of entropies with properties of branching, continuity, and weighted additivity. We deliberately do not assume that the entropies are symmetric. The resulting entropies are generalizations of the entropies of degree α, including the Shannon entropy as the case α = 1. Such “weighted” entropies have potential applications to the “utility of gambling” problem.
The functional equation to which the title refers is: F(x,y) + F(xy,z) = F(x,yz) + F(y,z), where x, y and z are in a commutative semigroup S and F: S x S --> X with (X,+) a divisible abelian group (Divisibility means that for any y belonging to X and natural number n there exists a (unique) solution x belonging to X to nx = y).
The functional equation is solved for general solution. The result is then applied to show that the three functional equations , and are equivalent. Finally, twice differentiable solution functions of the functional equation are determined.
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