Difference between revisions of "Schur's Inequality"
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=== Proof === | === Proof === | ||
− | [[WLOG]], let <math>{a \geq b \geq c}</math>. Note that <math>a^r(a-b)(a-c)+b^r(b-a)(b-c)</math> <math>= a^r(a-b)(a-c)-b^r(a-b)(b-c) = (a-b)(a^r(a-c)-b^r(b-c))</math>. Clearly, <math>a^r \geq b^r \geq 0</math>, and <math>a-c \geq b-c \geq 0</math>. Thus, <math>(a-b)(a^r(a-c)-b^r(b-c)) \geq 0 \ | + | [[WLOG]], let <math>{a \geq b \geq c}</math>. Note that <math>a^r(a-b)(a-c)+b^r(b-a)(b-c)</math> <math>= a^r(a-b)(a-c)-b^r(a-b)(b-c) = (a-b)(a^r(a-c)-b^r(b-c))</math>. Clearly, <math>a^r \geq b^r \geq 0</math>, and <math>a-c \geq b-c \geq 0</math>. Thus, <math>(a-b)(a^r(a-c)-b^r(b-c)) \geq 0 \implies a^r(a-b)(a-c)+b^r(b-a)(b-c) \geq 0</math>. However, <math>c^r(c-a)(c-b) \geq 0</math>, and thus the proof is complete. |
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=== Generalized Form === | === Generalized Form === | ||
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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. |
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.