On a generalization of sum form functional equation. V (Q1061954)

[For part IV see Indian J. Math. 22, 257-262 (1980; Zbl 0509.39012).] The following functional equation is considered which is connected with additive measures such as the Shannon entropy, the inaccuracy, the directed divergence, and also with some non-additive ones as the entropy of degree \(\beta\) etc., \[ \sum^{n}_{i=1} \sum^{m}_{j=1} f(x_ i y_ j, u_ i v_ j) = \sum^{n}_{i=1} x^2_ i\;cdot \sum^{m}_{j=1}f(y_ j,v_ j) + \sum^{m}_{j=1} y_ j^{\beta} \cdot \sum^{n}_{i=1}f(x_ i,u_ i), \tag{1} \] for \(U,X\in \Gamma_ n\), \(V,Y\in \Gamma_ m\), \(\alpha\), \(\beta\) non- zero reals. These equations were solved before, when the equations hold for all pairs \(m\), \(n\) or for some particular pairs (2.3), etc. Herein is obtained all the ''measurable'' solutions of (1) holding for some (arbitrary but) fixed pair m,n\(\geq 3:\) Let \(f:J\to\mathbb{R}\) be measurable in each variable. Then f is a solution of (1) for some arbitrary but fixed m,n (\(\geq3\)) if and only if \[ f(x,y) = \begin{cases} a(x^\alpha-x^\beta) + b(x-y),\quad& \alpha\neq\beta \\ ax^\alpha \log x^\alpha + bx \log y + c(x-y),\quad& \alpha =\beta, \quad \alpha\neq1 \text{ for } (x,y)\in J \\ ax\log x + bx\log y + cy\log y + Ax + By + D,\quad& \alpha=\beta=1 \end{cases} \] where \(a\), \(b\), \(c\), \(A\), \(B\) and \(D\) are constants satisfying \(A+B=(mn- m-n)D\).