Difference between revisions of "Schur's Inequality"

Latest revision as of 15:57, 29 December 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 $a,b,c \in \mathbb{R}$ 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: $a^3+b^3+c^3+3abc \ge a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b$

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

Proof

Without loss of Generality, let ${a\ge b\ge c}$ . 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, $a^r\ge b^r \ge 0$ , and $a-c \geq b-c \geq 0$ . Thus, $(a-b)(a^r(a-c)-b^r(b-c)) \geq 0 \implies a^r(a-b)(a-c)+b^r(b-a)(b-c) \geq 0$ . 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

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.

@@ Line 1: / Line 1: @@
-'''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''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur.
-<math>{a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0}</math>
+== Theorem ==
+Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>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>.
+<cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>
-=== Other forms ===
+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 <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>
+=== Common Cases ===
+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:
-<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>
+<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>
 === 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.
+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 ===
+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>
-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,
+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>.
-<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>
+== References ==
+* Mildorf, Thomas; ''Olympiad Inequalities'';  January 20, 2006;  <http://artofproblemsolving.com/articles/files/MildorfInequalities.pdf>
-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 ===
+* Vornicu, Valentin;  ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania.
-* Mildorf, Thomas; ''Olympiad Inequalities'';  January 20, 2006;  <http://www.mit.edu/~tmildorf/Inequalities.pdf>
+==See Also==
+* [[Olympiad Mathematics]]
+* [[Inequalities]]
+* [[Number Theory]]
-* Valentin, Vornicu;  ''Olimpiada de Matematica... de la provocare la experienta''; GIL Publishing House; Zalau, Romania.
+[[Category:Algebra]]
+[[Category:Inequalities]]

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