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

Authors:

Prem Nath Dhiraj Kumar Singh

Institution:

(1) Department of Mathematics, University of Delhi, Delhi, 110007, India

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.

Keywords:

sum form functional equation additive function multiplicative function

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