A generalization of entropy equation: homogeneous entropies

Shannon's entropy has been characterized in several ways. Kaminski and Mikusihski [5] simplified Fadeev's [3] approach by considering, what they called the entropy equation: (1.1) H(x, y, z) = H(x + y, 0, z) + H(x, y, 0) , (x ^ 0 , y ^ 0 , z > 0 , xy + yz + zx > 0) . A general continuous symmetric solution of (1.1) given by (1.2) H(x, y, z) = (x + y + z) log (x + y + z) — x log x y log y — z log z , was obtained by Kaminski and Mikusinski [5] under homogeneity of degree 1, viz. (1.3) H(Xx, Xy, Xz) = X H(x, y, z), X > 0 . Aczel [ l ] solved (1.1) under weaker regularity conditions. Sharma and Singh [6] relaxed (1.3) in two different ways: (i) by considering homogeneity of degree /?, given by (1.4) H(Xx, Xy, Xz) = X H(x, y, z), X > 0 , p> 0, j S + 1 , and (ii) by 'bi-homogeneity' of degree (a, 0) given by (1.5) H(Xx, Xy, Xz) = A X Ha(x, y, z) + B X" Hp(x, y, z), X> 0, a* p, P >0, a * 1 , j S * 1 , where A and B are arbitrary constants, and obtained the generalized solutions of