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Difference between revisions of "Schur's Inequality"

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The four equality cases occur when <math>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.
 
The four equality cases occur when <math>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.
  
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=== Other forms ===
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=== Common Cases ===
  
 
The <math>r=1</math> case yields the well-known inequality:<math>a^3+b^3+c^3+3abc \geq a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b</math>
 
The <math>r=1</math> case yields the well-known inequality:<math>a^3+b^3+c^3+3abc \geq a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b</math>
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When <math>r=2</math>, an equivalent form is:
 
When <math>r=2</math>, an equivalent form is:
 
<math>a^4+b^4+c^4+abc(a+b+c) \geq a^3 b+a^3 c+b^3 a+b^3 c+c^3 a+c^3 b</math>
 
<math>a^4+b^4+c^4+abc(a+b+c) \geq a^3 b+a^3 c+b^3 a+b^3 c+c^3 a+c^3 b</math>
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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) = 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 \rightarrow 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]], let <math>{a \geq b \geq c}</math>.  Note that <math>a^r(a-b)(a-c)+b^r(b-a)(b-c) = 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 \rightarrow 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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The standard form of Schur's is the case of this inequality where <math>x=a,\ y=b,\ z=c,\ k=1,\ f(m)=m^r</math>.
 
The standard form of Schur's is the case of this inequality where <math>x=a,\ y=b,\ z=c,\ k=1,\ f(m)=m^r</math>.
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=== References ===
 
=== References ===

Revision as of 21:48, 21 June 2006

Schur's Inequality states that for all non-negative $a,b,c \in \mathbb{R}$ and $r>0$:

${a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0}$

The four equality cases occur when $a=b=c$ or when two of $a,b,c$ are equal and the third is ${0}$.


Common Cases

The $r=1$ case yields the well-known inequality:$a^3+b^3+c^3+3abc \geq a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b$

When $r=2$, an equivalent form is: $a^4+b^4+c^4+abc(a+b+c) \geq a^3 b+a^3 c+b^3 a+b^3 c+c^3 a+c^3 b$


Proof

WLOG, let ${a \geq b \geq c}$. Note that $a^r(a-b)(a-c)+b^r(b-a)(b-c) = a^r(a-b)(a-c)-b^r(a-b)(b-c) = (a-b)(a^r(a-c)-b^r(b-c))$. Clearly, $a^r \geq b^r \geq 0$, and $a-c \geq b-c \geq 0$. Thus, $(a-b)(a^r(a-c)-b^r(b-c)) \geq 0 \rightarrow a^r(a-b)(a-c)+b^r(b-a)(b-c) \geq 0$. However, $c^r(c-a)(c-b) \geq 0$, 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 $a,b,c,x,y,z \in \mathbb{R}$, where ${a \geq b \geq c}$, and either $x \geq y \geq z$ or $z \geq y \geq x$. Let $k \in \mathbb{Z}^{+}$, and let $f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}$ be either convex or monotonic. Then,

${f(x)(a-b)^k(a-c)^k+f(y)(b-a)^k(b-c)^k+f(z)(c-a)^k(c-b)^k \geq 0}$

The standard form of Schur's is the case of this inequality where $x=a,\ y=b,\ z=c,\ k=1,\ f(m)=m^r$.


References

  • Valentin, Vornicu; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.
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