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Entropy of Degree s

The following characterization of the measure (3.3) is due to Havrda and Charvát (1967) .

A function will be said structural entropy if

(i) is continuous in the region for .
(ii) (iii)  (iv)   Axioms (i)-(iv) determine the structural entropy given by (3.3)

In the Rényi's case, Daróczy (1963)  restructured the axioms, while in the Havrda and Charvt's case, Daróczy (1970)  presented an alternative way to characterize the entropy of degree . This is as follows:

Let   be a real valued function satisfying: where  is a function such that for with , then (3.16)

and is as given by (3.3).

The function given by (3.16) is famous as "information function of degree ".

An alternative way to characterize the entropy of degree is following the Chaundy and McLeod's (1961)  approach. This can be seen in Sharma and Taneja (1975) , and is as follows:

Let  be a real valued function satisfying for all , and where is real valued continuous function with . Then is given by (3.3).

Some alternative approaches to characterize the measure (3.51) can be seen in Aczél and Daróczy (1975) , Mathai and Rathie (1975)  and Taneja (1979) .

It is worth emphasizing here that the measure (3.3), or better  satisfies some extra properties given below.

Property 3.18. We can write where Property 3.19. We can write Property 3.20. Let , then

(i) (ii) (iii) (iv) (v) Property 3.21. Let with . Then for all rationals Property 3.22. For all  , , we have

(i)  (ii) Property 3.23. For all , we have  for all .

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