## On a multiplicative type sum form functional equation and its role in information theory.(English)Zbl 1164.39330

Summary: We obtain all possible general solutions of the sum form functional equations \begin{aligned} \sum _{i=1}^{k}\sum _{j=1}^{\ell }f(p_iq_j)=&\sum _{i=1}^{k}g(p_i) \sum _{j=1}^{\ell }h(q_j)\qquad \text{and} \\ \sum _{i=1}^{k}\sum _{j=1}^{\ell }F(p_iq_j)=&\sum _{i=1}^{k} G(p_i)+\sum _{j=1}^{\ell }H(q_j)+ \lambda \sum _{i=1}^{k}G(p_i)\sum _{j=1}^{\ell }H(q_j) \end{aligned} valid for all complete probability distributions $$(p_1,\ldots ,p_k)$$, $$(q_1,\ldots ,q_\ell )$$, $$k\geq 3$$, $$\ell \geq 3$$ fixed integers; $$\lambda \in \mathbb R$$, $$\lambda \neq 0$$ and $$F$$, $$G$$, $$H$$, $$f$$, $$g$$, $$h$$  are real valued mappings each having the domain $$I=[0,1]$$, the closed unit interval.

### MSC:

 39B22 Functional equations for real functions 94A15 Information theory (general)
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### References:

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