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
Property 4.9. (Generalized data processing inequality). We have
where and are the probability distributions given by
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
for all , where and are determined by the equations: