Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 709

whenever x 2 is not a permutation of x 1 . ♦

Clearly f ( x ) is Schur-convex if and only if −f ( x ) is Schur-concave. When we
say that f ( x ) is Schur-convex in a subset S of real vectors, we mean that the
argument x is constrained to be in the domain

S .

Example 21.3: Schur-convex functions

Perhaps the simplest example of a Schur-convex function is

f ( x )=max
k

{ x k } = x [0] . (21 . 30)

If x y then clearly x [0] ≥ y [0] , which proves that f ( x ) ≥ f ( y ) indeed. For

a less trivial example consider

f ( x )= 1
x 0

1
x 1

+

We will show later (see remarks around Eq. (21.38)) that this is Schur-

convex in 0 <x k < ∞ . Here we will evaluate this function for the two

vectors

x 1 =[4 2] , x 2 =[3 3] .

Clearly x 1 x 2 .Wehave

f ( x 1 )= 1

4 + 1

f ( x 2 )= 1

3 + 1 2 =0 . 75

and

3 ≈

0 . 667 ,

so that f ( x 1 ) >f ( x 2 ) indeed.

21.4 Examples of Schur-convex functions

There are many beautiful examples of Schur-concave/convex functions which
arise in optimization problems in signal processing and communications. Some
of these examples follow from the theorems to be presented in this section.
The first theorem shows the relation between convex functions and Schur-
convex functions (see p. 64 of Marshall and Olkin [1979]). In what follows, the
symbol I denotes an interval on the real line (e.g., 0 . 5 <x ≤∞ , − 2 ≤ x ≤ 3,

P

and so forth), and I denotes the set of P -vectors whose components x k are in

I .

Theorem 21.3. From convex to Schur-convex functions. Let g ( x ) be convex
on some interval I of the real line. Then the function f ( x )= P− 1
k =0

♠

g ( x k )is

P .

Schur-convex on I ♦