# Difference between revisions of "Schur's Inequality"

m |
m (proofreading) |
||

Line 3: | Line 3: | ||

<math>{a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0}</math> | <math>{a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0}</math> | ||

− | The four equality cases occur when <math>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>. | + | The four equality cases occur when <math>a=b=c</math>, or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>. |

## Revision as of 12:26, 23 June 2006

**Schur's Inequality** states that for all non-negative and :

The four equality cases occur when , or when two of are equal and the third is .

### Common Cases

The case yields the well-known inequality:

When , an equivalent form is:

### Proof

WLOG, let . Note that . Clearly, , and . Thus, . However, , and thus the proof is complete.

### Generalized Form

It has been shown by Valentin Vornicu that a more general form of Schur's Inequality exists. Consider , where , and either or . Let , and let be either convex or monotonic. Then,

The standard form of Schur's is the case of this inequality where .

### References

- Mildorf, Thomas;
*Olympiad Inequalities*; January 20, 2006; <http://www.mit.edu/~tmildorf/Inequalities.pdf>

- Valentin, Vornicu;
*Olimpiada de Matematica... de la provocare la experienta*; GIL Publishing House; Zalau, Romania.