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Properties of M-Dimensional Unified (r,s)-Divergence Measures

In this subsection we shall give some properties of dimensional unified measures  and ( and 3).

Property 5.1. The measures ( and 2), ( and 3) and ( and 3) are nonnegative for all and any are zero iff . The nonnegativity of follows under the conditions when either or with .

Property 5.2. For all , we have  and Property 5.3. For all , we have and Property 5.4. For all  and 2, we have  Property 5.5. The measures ( =1 and 2), ( =1,2 and 3) and ( =1,2 and 3) are increasing functions of ( fixed) and of ( fixed). In particular, when , the result still holds.

Property 5.6. For  (0, ),  ( ), =1 and 2, the measures   and are Schur-convex functions of i.e., implies  and Property 5.7. (Generalized data processing inequalities). For  (0, ),  ( ), =1 and 2, we have  and where with B= , i=1,2,...,n; j=1,2,...,m being a stochastic matrix such that =1, i=1,2,...,n.
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil