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
Property 5.3. For all , we have
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
Property 5.7. (Generalized data processing inequalities). For (0,), (), =1 and 2, we have