On a sum form functional equation
The functional equations considered are (1) \(I(pq,vw)=g(q,w)I(p,v)+f(p,v)I(q,w)\) for all p,q\(\in [0,1]\), v,w\(\in (0,\infty)\), and (2) \(\sum_{k}\sum_{j}I(p_ kq_ j,v_ kw_ j)=\sum_{k}\sum_{j}[g(q_ j,w_ j)I(p_ k,v_ k)+f(p_ k,v_ k)I(p_ k,v_ k)I(q_ j,w_ j)]\) for all complete probability distributions \((p_ 1,...,p_ n)\), \((q_ 1,...,q_ m)\) of (fixed) length n and m, respectively, and all n-tuples [m-tuples] \((v_ 1,...,v_ n)\) \([(w_ 1,...,w_ m)]\) of positive real numbers. In both equations, I, f, g map [0,1]\(\times (0,\infty)\) into the reals. The second equation arises in characterizing entropies with a generalized additivity property. The author first solves (1) (an extension to 2 dimensions of known 1-dimensional results), then uses that result to find the measurable solutions of (2) for fixed pairs (n,m) with n,m\(\geq 3\). This generalizes a result of \textit{PL. Kannappan} [Util. Math. 22, 141-147 (1982; Zbl 0509.39013)], in which f and g are known functions from the start.
scientific article; zbMATH DE number 3921166