Convexity, Schur convexity, and majorization theory - Signal Processing and Optimization for Transceiver Systems - page 729

Table 21.4. Schur-convex functions: examples

Set D is x 0 ≥ x 1 ≥ ... ≥ x P− 1 , set D + is x 0 ≥ x 1 ≥ ... ≥ x P− 1 ≥ 0 , and set D ++ is

x 0 ≥ x 1 ≥ ... ≥ x P− 1 > 0 . It is assumed throughout that 0 ≤ a 0 ≤ a 1 ≤ ... ≤ a P− 1 .

Here are some examples of Schur-convex functions.
1. exp( P− 1
k

1 /x k )for x k > 0 .

=0

2. P− 1
k

− P− 1
k

1 / √ 2and 1 / √ 2 .

e −x k in e −x k in | x k |≥ | x k |≤

=0 =0

3. P− 1
k

erfc(1 / √ x k )in0 <x k ≤ 2 / 3 .

=0

4. max k {x k } for any real x .
5. P− 1
k

a k /x k ( p> 0) in D ++
6. Examples of above: P− 1
k

=0

a k / √ x k , P− 1
k

a k /x k , P− 1
k

a k /x k in D +

=0 =0 =0

7. P− 1
k

a k / (1 + x k )in D +

=0

8. P− 1
k

a k e −αx k ( α> 0) in D +

=0

9. − P− 1
k

a k ln x k in D ++

=0

− P− 1
k

x a k
k

10. in D ++

=0

11. P− 1
k

a k / (1 + x k ) 2 is Schur convex in x 0 ≥ x 1 ≥ ...≥ x P− 1 ≥ 1 / √ 3 .

=0

− P− 1
k

12. a k x k in D .

=0

13. P− 1
k

a k (1 − x k ) /x k in D ++

=0

14. P− 1
k

a k ln[(1 − x k ) /x k ]in D ++ if x k ≤ 0 . 5 .

=0

15. P− 1
k

[(1 − x k ) /x k ] a k in D ++ if x k ≤ 0 . 5 .

=0

Note: All functions, arguments, and constants are real-valued .