Introduction to majorization - Statistical Signal Processing of Complex-Valued Data - page 292

symmetric and convex, then it is also Schur-convex. The reverse conclusion is not
valid.

Example A3.3. Consider again the function g ( x ) from Example A3.2 , which is of the
form ( A3.13 ) with

x i

x i .
It can be checked that h ( x i ) is convex by showing that its second derivative is nonnegative.
Therefore, g ( x ) is Schur-convex. This is exactly what we have done in Example A3.2 ,
thus implicitly deriving the special rule for functions of the form ( A3.13 ).

h ( x i ) =

1 −

Example A3.4. Consider a discrete random variable X that takes on n values with proba-
bilities p 1 , p n ] T . The entropy

p 2 ,..., p n , which we collect in the vector p = [ p 1 , p 2 ,...,

of X is given by

n

H ( X ) =− p i log p i .

i = 1

Since h ( x ) x log x is strictly convex, entropy is strictly Schur-concave. Using the result

from Example A3.1 , it follows that minimum entropy is achieved when probabilities are
“most unequal”, i.e., p

=

0] T or any permutation thereof. Maximum entropy
is achieved when all probabilities are equal, i.e., p

= [1 , 0 ,...,

n ] T .

= [1 / n , 1 / n ,..., 1 /

A3.2.2 Functions defined on D

We know from Result A3.3 that Schur-convex functions are necessarily symmetric if they
are defined on IR n . However, it is not required that functions defined on the set of ordered
n -tuples

D ={ x : x 1 ≥ x 2 ≥···≥ x n } be symmetric in order to be Schur-convex.

D → D

Result A3.4. Let g : IR be continuous on and continuously differentiable on the

interior of D . Then g is Schur-convex if and only if

∂
∂

∂

x i g ( x )

≥

x i + 1 g ( x )

, i = 1 ,..., n − 1 , (A3.16)

∂

for all x in the interior of D . For functions of the form

n

g ( x ) = h i ( x i ) , (A3.17)

i = 1

where each h i :IR → IR is differentiable, this condition simplifies to

h i ( x i ) h i + 1 ( x i + 1 ) ≥ , ∀ x i ≥ x i + 1 , i = 1 ,..., n − 1 . (A3.18)