Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 712

dg ( x )
dx

2. ≤ 0( g ( x ) monotone non increasing);

d 2 g

( x )

3. ≥ 0( g ( x ) convex).

dx 2

Then f ( x ) is Schur-convex on D (i.e., for x 0 ≥ x 1 ≥ ... ≥ x P− 1 ). ♦

Proof. Throughout the proof remember that the condition x 0 ≥ x 1 ≥ ...≥
x P− 1 is assumed. The notation dg ( x k ) /dx denotes the derivative of g ( x )
evaluated at x k . Since d 2 g ( x ) /dx 2 ≥ 0, we have

dg ( x 0 )
dx ≥

dg ( x 1 )
dx ≥ ...≥

dg ( x P − 1 )
dx

.

Using dg ( x ) /dx ≤ 0 we therefore get

dg ( x 0 )
dx ≤−

dg ( x 1 )
dx ≤

dg ( x P − 1 )
dx

0 ≤− ... ≤− .

We now combine this with the first condition of the corollary:

0 ≤ a 0 ≤ a 1 ≤ ... ≤ a P− 1 .

Since a k and dg ( x k ) /dx are non-negative we conclude from the preceding

two sets of inequalities that

−

dg ( x 0 )
dx ≤−

dg ( x 1 )
dx ≤

dg ( x P − 1 )
dx

0 ≤− a 0 a 1 ... ≤− a P− 1

which is equivalent to

dg ( x 0 )
dx ≥

dg ( x 1 )
dx ≥

dg ( x P − 1 )
dx

0 ≥ a 0 a 1 ... ≥ a P− 1

The function (21.36) therefore has the form (21.35) (just set g k ( x )= a k g ( x ))
and satisfies all the conditions of Theorem 21.4. It is therefore a Schur-
convex function in D .

For example, let

g ( x )= 1

x p ,

p> 0 .

Then, for x> 0 ,

< 0and d 2 g ( x )
dx 2

dg ( x )
dx

= −

p
x p +1

= p ( p +1)
x p +2

> 0 .

Applying Corollay 21.1 it therefore follows that

P− 1

a k
x k

f ( x )= , p> 0 (21 . 37)

k =0