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Entropy of Order r and
Degree s
Sharma and Mittal (1975) [90] presented an
axiomatic characterization of entropy of order
and degree.
It is based on the Rényi's approach, where the additivity property
has been changed (generally referred as nonadditivity).
Let
be a real valued continuous function satisfying
where g is a strictly monotonic continuous function, and
is the selfinformation of an event of a probability distribution
satisfying:

(i)
is a continuous function
in (0,1].

(ii)

(iii)
Then



(3.20) 
and



(3.21) 
Van der Pyl (1977) [118]restructured the
above axiomatic system and considered as follows:
Let
be a real valued continuous function satisfying the following:

(i)
is a symmetric function of its arguments.

(ii)
is continuous in (0,1].

(iii)

(iv) There is a sequence
such that
for all ,
and.

(v) There exists a strictly monotonic continuous function g such that
Then the above set of axioms lead to the measures (3.20) and (3.21).
Picard (1979) [77] extended the above
set of axioms by introducing the idea of weights or preferences and came
up with the weighted entropies given in section 3.6.2.
The measure (3.21) can be characterized in a much more simplified form
given as follows:
Let
be a real valued continuous function satisfying the following axioms:

(i)
where f is a continuous function defined on [0,1].

(ii)
for ,
and .

(iii)
Then
that is same as (3.21) for
and ,
with .
21062001
Inder Jeet Taneja
Departamento de Matemática  UFSC
88.040900 Florianópolis, SC  Brazil