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
WLOG(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.