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## Entropy of Degree s

The following characterization of the measure (3.3) is due to Havrda and Charvát (1967) [46].

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) [31] restructured the axioms, while in the Havrda and Charvt's case, Daróczy (1970) [33] 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) [27] approach. This can be seen in Sharma and Taneja (1975) [92], 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) [2], Mathai and Rathie (1975) [71] and Taneja (1979) [99].

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