# Difference between revisions of "Schur's Inequality"

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It has been shown by [[Valentin Vornicu]] that a more general form of Schur's Inequality exists. Consider <math>a,b,c,x,y,z \in \mathbb{R}</math>, where <math>{a \geq b \geq c}</math>, and either <math>x \geq y \geq z</math> or <math>z \geq y \geq x</math>. Let <math>k \in \mathbb{Z}^{+}</math>, and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}</math> be either convex or monotonic. Then, | It has been shown by [[Valentin Vornicu]] that a more general form of Schur's Inequality exists. Consider <math>a,b,c,x,y,z \in \mathbb{R}</math>, where <math>{a \geq b \geq c}</math>, and either <math>x \geq y \geq z</math> or <math>z \geq y \geq x</math>. Let <math>k \in \mathbb{Z}^{+}</math>, and let <math>f:\mathbb{R} \rightarrow \mathbb{R}_{0}^{+}</math> be either convex or monotonic. Then, | ||

− | <math>{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}</math> | + | <math>{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}</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>. | 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 14:10, 27 October 2007

**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://www.mit.edu/~tmildorf/Inequalities.pdf>

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