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On characterization of directed divergence of type \(\beta\) through information equation. (English) Zbl 0548.94009
The authors are trying to solve the equations \[ (1.3)\quad I\begin{pmatrix} x,y,z \\ l,m,n \end{pmatrix} = I\begin{pmatrix} x+y,0,z \\ l+m,0,n \end{pmatrix} + I\begin{pmatrix} x,y,0 \\ l,m,0 \end{pmatrix} \] and \[ (1.5)\quad I\begin{pmatrix} \lambda x,\lambda y,\lambda z \\ \mu l,\mu m,\mu n \end{pmatrix} = \lambda^{\beta}\mu^{1-\beta} I\begin{pmatrix} x,y,z \\ l,m,n \end{pmatrix}, \] \(\lambda,\mu >0\), \(\beta >0\) and \(\neq 1\) on \((*)\quad D^ 2=\{(x,y,z;l,m,n):x,y,z,l,m,n\geq 0\) with \(xy+yz+zx>0\), \(lm+mn+nl>0\}\) under the assumption of symmetry of I.
Reviewer’s remarks. The authors are not clear about the domain and the notations they are using, for example in (2.13), the authors find the value of \(I\begin{pmatrix} x,0,0 \\ l,0,0 \end{pmatrix}\) which does not satisfy (*). Also the authors define f (in 2.14) on \([0,1]\times [0,1]\) in terms of I and use \(f(1,1)=f(0,0)\) (which involves \(I\begin{pmatrix} 0,1,0 \\ 0,2,0 \end{pmatrix}\) etc., which are not defined at all) to determine I. It is also not made clear, why I satisfying (1.3) and (1.5) should be of the form (2.5) which involves the function \(I^{\beta}_{n}\) satisfying the postulates 1 to 4. Thus, the authors have not determined all the solutions of (1.3) and (1.5) under symmetry. Further, the reviewer with P. N. Rathie has characterized \(I_ n^{\beta}\) satisfying the postulates 1 to 4 (even weaker in case of postulate 2) using a functional equation [Inf. Control 20, 38-45 (1972; Zbl 0231.94015)]. The reviewer (joint with Kamiński and Mikusiński) has characterized directed divergence, inaccuracy, generalized directed divergence by using equations of the form (1.3), (1.5) and (2.16) [A. Kamiński, the reviewer and J. Mikusiński, Ann. Pol. Math. 36, 101-110 (1979; Zbl 0404.94004); the reviewer and A. Kamiński, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 25, 925-928 (1977; Zbl 0365.94028) and Ann. Pol. Math. 38, 289-294 (1980; Zbl 0449.94008), which are all missing as references].
Reviewer: Pl.Kannappan

MSC:
94A17 Measures of information, entropy
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References:
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