Let
beprobability
distributions. Let .
It is well known that the Shannon's entropy satisfy the following
inequality:

(5.1) 
with equality iff .
Also due to property 1.14, we have

(5.2) 
for each k=1,2,...,M with equality iff .
We can easily check that

(5.3) 
Multiplying both sides of (5.2) by , summing over k=1,2,...,M and using (5.3), we have

(5.4) 
with equality iff .
From (5.1) and (5.4), we have

(5.5) 
with equality iff .
The above inequality (5.5) admits the following three nonnegative differences given by

(5.6) 

(5.7) 
and

(5.8) 
From (5.6), (5.7) and (5.8), we conclude that

(5.9) 
From (5.9), we have the following inequalities:

(5.10) 
and

(5.11) 
The measures (5.6), (5.7) and (5.8) in particular reduce to (2.6) and (2.7) respectively, when M=2, and with a multiplicative constant.
Note 5.1.