On the general solution of a functional equation connected to sum form information measures on open domain. III (Q1093104)

[For part I see Publ. Inst. Math., Nouv. Ser. 40(54), 57-62 (1986; Zbl 0619.39009).] This paper is devoted to the study of a functional equation connected with the characterization of weighted entropy and weighted entropy of degree \(\beta\). Here, we find the general solution of the functional equation \[ \sum^{n}_{i=1}\sum^{m}_{j=1}f(p_ iq_ j,u_ iv_ j)=\sum^{n}_{i=1}p_ i^{\alpha}u_ i\cdot \sum^{m}_{j=1}f(q_ i,v_ j)+\sum^{m}_{j=1}q_ j^{\beta}v_ j\cdot \sum^{n}_{j=1}f(p_ i,u_ i) \] on an open domain, without using 0-probability and 1-probability.