A sum form functional equation on a closed domain and its role in information theory

This paper is devoted to finding the general solutions of the functional equation $\sumin \sumjm h(p_iq_j)=\sumin h(p_i)+\sumjm k_j(q_j)+\lambda\sumin h(p_i)\sumjm k_j(q_j)$validforallcompleteprobabilitydistributions$(p_1,\ldots,p_n)$,$(q_1,\ldots,q_m)$,$0\le p_i\le 1$,$0\le q_j\le 1$,$i=1,\ldots,n$;$j=1,\ldots,m$,$\sumin p_i=1$,$\sumjm q_j=1$;$n\ge 3$,$m\ge 3$ fixed integers; $\lambda \in \text{RR}$, $\lambda \mathrm{\ne }0$ and the mappings $h:I\to \text{RR}$, $k_j:I\to \text{RR}$, $j=1,\dots ,m$; $I=[0,1]$, $\text{RR}$ denoting the set of all real numbers. A special case of the above functional equation was treated earlier by L. Losonczi and Gy. Maksa.