# Schur's Inequality

**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

Without loss of Generality, 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.