Additive information measures with preference of order \(\beta\) (Q1096585)

The authors present all continuous functions \(f,g,h: [0,1]\times [0,\infty]\to [0,\infty)\) satisfying the functional equation \[ (1)\quad \sum^{n}_{i=1}\sum^{m}_{j=1}f(p_ iq_ j,u_ iv_ j)=\sum^{n}_{i=1}\sum^{m}_{j=1}(v_ jq_ j)^{\beta}g(p_ i,u_ i)+\sum^{n}_{i=1}\sum^{m}_{j=1}(u_ ip_ i)^{\beta}h(q_ j,v_ j),\quad \beta >0 \] and give a characterization of certain weighted-additive information measures. Remark: A more general result where in (1) the expressions \((v_ jq_ j)^{\beta}\) and \((u_ ip_ i)^{\beta}\) will be replaced by two different multiplicative functions \(M_ 1(q_ j,v_ j)\) and \(M_ 2(p_ iu_ i)\), and where f, g, h are arbitrary functions, is due to the reviewer and will appear elsewhere.