# Difference between revisions of "Schur's Inequality"

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'''Schur's Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. | '''Schur's Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. | ||

− | + | == 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>: | ||

− | < | + | <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath> |

The four [[equality condition | 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 condition | 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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=== Common Cases === | === Common Cases === | ||

− | + | The <math>r=1</math> case yields the well-known inequality: | |

− | The <math>r=1</math> case yields the well-known inequality:< | + | <cmath>a^3+b^3+c^3+3abc \ge a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b</cmath> |

When <math>r=2</math>, an equivalent form is: | When <math>r=2</math>, an equivalent form is: | ||

− | < | + | <cmath>a^4+b^4+c^4+abc(a+b+c) \ge a^3 b+a^3 c+b^3 a+b^3 c+c^3 a+c^3 b</cmath> |

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=== Proof === | === Proof === | ||

− | + | Without loss of Generality, let <math>{a\ge b\ge 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\ge b^r \ge 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 === | ||

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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, | ||

− | + | <cmath>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.</cmath> | |

− | < | ||

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>. | ||

== References == | == References == | ||

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* Mildorf, Thomas; ''Olympiad Inequalities''; January 20, 2006; <http://artofproblemsolving.com/articles/files/MildorfInequalities.pdf> | * Mildorf, Thomas; ''Olympiad Inequalities''; January 20, 2006; <http://artofproblemsolving.com/articles/files/MildorfInequalities.pdf> | ||

## Latest revision as of 18:19, 17 June 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

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.