On a functional equation related to some entropies in information theory

The general solutions of the functional equation Sigma(n)(i=1)Sigma(m)(j=1)f(p(i)q(i)) = Sigma(n)(i=1)g(p(i))Sigma(m)(j=1)h(q(j))+Sigma(n)(i=1)p(i)(alpha)Sigma(m)(j=1)f(q(j)) in which f, g, h are real-valued mappings each with domain I = [0,1]; (p(1),...,p(n))is an element of Gamma(n) ,(q(1),..., q(m))is an element of Gamma(m); n >= 3, m >= 3 being fixed integers; alpha > 0,alpha not equal 1,alpha is an element of R; have been obtained. Some of the solutions are related to entropies of degree alpha; entropies of degree (alpha, beta); and to Gini-Simpson index.