Functional equations in civil law (Q1818261)

The application of an Austrian law about keeping the relative values of shares in a shared property relatively stable in time, with a condition for permanence of arithmetic means added, leads to the system of functional equations \[ \begin{multlined} f_{jk}(\frac{p_1+r_1}{2},\ldots,\frac{p_n+r_n}{2};q_1,\ldots,q_m)= \\ =\frac{f_{jk}(p_1 ,\ldots,p_n;q_1,\ldots,q_m) + f_{jk}(r_1,\ldots,r_n; q_1,\ldots,q_m)}{2}\end{multlined} \] and \[ \sum_{k=1}^m f_{jk}(p_1,\ldots,p_n;q_1,\ldots,q_m)=p_j,\quad \sum_{j=1}^n f_{jk}(p_1,\ldots,p_n;q_1,\ldots,q_m)=q_k, \] where \(p_j>0\), \(r_j>0\), \(q_k>0\),\(\sum_{j=1}^n p_j=\sum_{j=1}^n r_j=\sum_{k=1}^m q_k=c\) (a positive constant). The author offers \(f_{jk}(p_1,\ldots,p_n;q_1,\ldots,q_m)=p_j q_k/c\) as general nonnegative-valued solutions \((j=1,\ldots,n;\: k=1,\ldots,m).\) Remark: In addition to the references quoted in Remark 1 of the paper regarding the Lemma, also the article by \textit{F. Radó} and \textit{J. A. Baker} [Aequationes Math. 32, 227-239 (1987; Zbl 0625.39007)] seems relevant.