On a multiplicative type sum form functional equation and its role in information theory

Abstract

In this paper, we obtain all possible general solutions of the sum form functional equations

$$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {f(p_i q_j )} } = \sum\limits_{i = 1}^k {g(p_i )} \sum\limits_{j = 1}^l {h(q_j )} $$

and

$$\sum\limits_{i = 1}^k {\sum\limits_{j = 1}^l {F(p_i q_j )} } = \sum\limits_{i = 1}^k {G(p_i ) + } \sum\limits_{j = 1}^l {H(q_j ) + \lambda } \sum\limits_{i = 1}^k {G(p_i )} \sum\limits_{j = 1}^l {H(q_j )} $$

valid for all complete probability distributions (p ₁, ..., p _k), (q ₁, ..., q _l ), k ≥ 3, l ≥ 3 fixed integers; λ ∈ ℝ, λ ≠ 0 and F, G, H, f, g, h are real valued mappings each having the domain I = [0, 1], the unit closed interval.