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Entropy of Kind t

To obtain an entropy of kind t, Arimoto (1971) [3] used a different approach. Let  be a real valued scalar function defined and nonnegative on (0,1] with a continuous derivative on (0,1] and . For all, let us define
 (3.19)
where the inf. is taken over all probability distributions such that .

The function  given by (3.19) satisfy some interesting properties (ref. Arimoto, 1971) [3]. These are summarized in the following property.

Property 3.24. We have

(i)  is a continuous and symmetric function with respect to its arguments  .
(ii) .

(iii)  is a concave function with respect to  in .

(iv) .

(v) If  is convex, then
(vi) In general
(vii) If  on (0,1), then (a)  (b)
(viii) If  is convex with  on (0,1], then
where .
Let
then from (3.19), one has

The function  is the entropy of kind . It arises as a particular case of (3.19). Some other examples of (3.19) can be seen in Arimoto (1971) [3]. Boeeke et al. (1980) [16] studied extensively the entropy of kind  by naming norm, considering . Some properties due to Behara and Chawla (1974) [9] are also worth emphasizing. Following Rényi's approach, the measure  can be characterized. It is not specified here, because the next section cover it as a particular case.

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil