Note on generalization of Shannon theorem and inequality

Recently, Santos obtained a generalized entropy using four assumptions which stared that an entropy must: (i) be a continuous function of the probabilities {p(i)}; (ii) be a monotonic increasing function of the number of states W, in the case of equiprobability; (iii) satisfy S-q(T) (A + B)/k = S-q(T) (A)/k + S-q(T) (B)/k + (1 - q) S-q(T) (A) S-q(T) (B)/k(2) (where A and B are two independent systems) and (iv) satisfy the relation S-q(T) ({p(i)}) = S-q(T) ({p(L), p(M)}) + p(L)(q) S-q(T) ({p(i)/p(L)}) + p(M)(q) S-q(T) ({p(i)/p(M)}), where p(L) + p(M) = 1 (p(L) = Sigma(i=1)(WL) p(i) and p(M) = Sigma(i=WL)(W) p(i)). Santos showed that the only function which satisfies all of these properties is the generalized Tsallis entropy. In this paper we perform a similar analysis and we obtain a family of entropies which are equivalent to the Tsallis entropy. We also discuss the Shannon inequality in the context of the generalized Tsallis entropy.