Difference between revisions of "Schur's Inequality"
m (Repaired dead link) |
(→Theorem) |
||
Line 2: | Line 2: | ||
− | == | + | == THEOREM == |
Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: | Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: | ||
Line 20: | Line 20: | ||
=== 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 \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. | + | [[WLOG(Without loss of Generality)]], 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. |
=== Generalized Form === | === Generalized Form === |
Revision as of 09:35, 23 April 2021
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.