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 .
The case yields the well-known inequality:
When , an equivalent form is:
WLOG, let . Note that . Clearly, , and . Thus, . However, , and thus the proof is complete.
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 .
- 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.