# Difference between revisions of "Schur's Inequality"

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== References == | == References == | ||

− | * Mildorf, Thomas; ''Olympiad Inequalities''; January 20, 2006; <http:// | + | * Mildorf, Thomas; ''Olympiad Inequalities''; January 20, 2006; <http://artofproblemsolving.com/articles/files/MildorfInequalities.pdf> |

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

## Latest revision as of 21:38, 11 September 2015

**Schur's Inequality** is an inequality that holds for positive numbers. It is named for Issai Schur.

## Theorem

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://artofproblemsolving.com/articles/files/MildorfInequalities.pdf>

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