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: