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## Properties of Unified (r,s)-Relative Information

The unified relative information given by (4.1) satisfies the following properties;

Property 4.1. (Continuity). is a continuous function of the pair  and is also continuous with respect to the parameters  and .

Property 4.2. (Symmetry). is a symmetric function of their arguments in the pair , i.e.,

where  is an arbitrary permutation of  to .

Property 4.3. (Expansibility). We can write

Property 4.4.(Nonadditivity). We have

for all  and .

Property 4.5. (Nonnegativity). with equality iff .

Property 4.6. (Monotonicity). is an increasing function of  ( fixed) and of ( fixed). In particular, when , the result still holds.

Property 4.7. (Inequalities among the measures). We have

(i)
(ii)
Property 4.8. (Convexity). is a convex function of the pair of probability distributions  for .

Property 4.9. (Generalized data processing inequality). We have

where  and  are the probability distributions given by

and
where  is a stochastic matrix such that.

Property 4.10. (Schur-convexity) is a Schur-convex function in the pair .

Property 4.11. For , we have

Property 4.12. (Order preserving) We have

(i) If , then

for all , where  and  are determined by the equations:

and
(ii) If , then

where

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