Geodesic
On a multiplicative type sum form functional equation and its role in information theory
Applications of Mathematics, Tome 51 (2006) no. 5, pp. 495-516

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In this paper, we obtain all possible general solutions of the sum form functional equations \[ \align \sum_{i=1}^{k}\sum_{j=1}^{\ell}f(p_iq_j)=\sum_{i=1}^{k}g(p_i) \sum_{j=1}^{\ell}h(q_j)\\ \text{and} \sum_{i=1}^{k}\sum_{j=1}^{\ell}F(p_iq_j)=\sum_{i=1}^{k} G(p_i)+\sum_{j=1}^{\ell}H(q_j)+ \lambda\sum_{i=1}^{k}G(p_i)\sum_{j=1}^{\ell}H(q_j) \endalign \] valid for all complete probability distributions $(p_1,\ldots ,p_k)$, $(q_1,\ldots ,q_\ell )$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in \mathbb{R}$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.
In this paper, we obtain all possible general solutions of the sum form functional equations \[ \align \sum_{i=1}^{k}\sum_{j=1}^{\ell}f(p_iq_j)=\sum_{i=1}^{k}g(p_i) \sum_{j=1}^{\ell}h(q_j)\\ \text{and} \sum_{i=1}^{k}\sum_{j=1}^{\ell}F(p_iq_j)=\sum_{i=1}^{k} G(p_i)+\sum_{j=1}^{\ell}H(q_j)+ \lambda\sum_{i=1}^{k}G(p_i)\sum_{j=1}^{\ell}H(q_j) \endalign \] valid for all complete probability distributions $(p_1,\ldots ,p_k)$, $(q_1,\ldots ,q_\ell )$, $k\ge 3$, $\ell \ge 3$ fixed integers; $\lambda \in \mathbb{R}$, $\lambda \ne 0$ and $F$, $G$, $H$, $f$, $g$, $h$ are real valued mappings each having the domain $I=[0,1]$, the unit closed interval.
DOI : 10.1007/s10492-006-0018-6
Classification : 39B22, 39B52, 39B82, 94A15
Keywords: sum form functional equation; additive function; multiplicative function 
Nath, Prem; Singh, Dhiraj Kumar. On a multiplicative type sum form functional equation and its role in information theory. Applications of Mathematics, Tome 51 (2006) no. 5, pp. 495-516. doi: 10.1007/s10492-006-0018-6
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