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| == you know what Air Jordan 5 Retro Australia ==
| | In [[mathematics]], '''Nesbitt's [[inequality (mathematics)|inequality]]''' is a special case of the [[Shapiro inequality]]. It states that for positive real numbers ''a'', ''b'' and ''c'' we have: |
| | :<math>\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}.</math> |
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| | ==Proof== |
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| | ===First proof=== |
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| | Starting from Nesbitt's inequality(1903) |
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| | :<math> \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}</math> |
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| | we transform the left hand side: |
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| | :<math>\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}-3\geq\frac{3}{2}.</math> |
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| | Now this can be transformed into: |
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| | :<math>((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)\geq 9.</math> |
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| | Division by 3 and the right factor yields: |
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| | :<math>\frac{(a+b)+(a+c)+(b+c)}{3}\geq\frac{3}{\frac{1}{a+b}+\frac{1}{a+c}+ \frac{1}{b+c}}.</math> |
| </ul>
| | Now on the left we have the [[arithmetic mean]] and on the right the [[harmonic mean]], so this inequality is true. |
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| == Rizwan Shah Michael Kors Jet Set Travel Tote ==
| | We might also want to try to use GM for three variables. |
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| | ===Second proof=== |
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| | Suppose <math> a \ge b \ge c </math>, we have that |
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| | :<math>\frac 1 {b+c} \ge \frac 1 {a+c} \ge \frac 1 {a+b} </math> |
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| | define |
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| | :<math>\vec x = (a, b, c) </math> |
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| | :<math>\vec y = \left(\frac 1 {b+c} , \frac 1 {a+c} , \frac 1 {a+b}\right) </math> |
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| | The scalar product of the two sequences is maximum because of the [[Rearrangement inequality]] if they are arranged the same way, call <math>\vec y_1 </math> and <math>\vec y_2</math> the vector <math>\vec y</math> shifted by one and by two, we have: |
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| | :<math>\vec x \cdot \vec y \ge \vec x \cdot \vec y_1</math> |
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| | :<math>\vec x \cdot \vec y \ge \vec x \cdot \vec y_2</math> |
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| </ul>
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| == "I thought Louis Vuitton Sale ==
| | Addition yields Nesbitt's inequality. |
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| | === Third proof === |
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| | The following identity is true for all <math>a,b,c:</math> |
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| == . Louis Vuitton Sale Melbourne == | | :<math>\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} = \frac{3}{2} + \frac{1}{2} \left(\frac{(a-b)^2}{(a+c)(b+c)}+\frac{(a-c)^2}{(a+b)(b+c)}+\frac{(b-c)^2}{(a+b)(a+c)}\right) |
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| | </math> |
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| | This clearly proves that the left side is no less than <math>\frac{3}{2}</math> for positive a,b and c. |
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| == Western WoodPewee Pandora Bracelet Charms ==
| | Note: every rational inequality can be solved by transforming it to the appropriate identity, see [[Hilbert's seventeenth problem]]. |
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| | === Fourth proof === |
|
| | Starting from Nesbitt's inequality(1903) |
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| | :<math> \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}</math> |
|
| | We add <math>3</math> to both sides. |
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| | :<math>\frac{a+b+c}{b+c}+\frac{a+b+c}{a+c}+\frac{a+b+c}{a+b}\geq\frac{3}{2}+3</math> |
|
| | Now this can be transformed into: |
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| | :<math>(a+b+c)\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\geq\frac{9}{2}</math> |
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| | Multiply by <math>2</math> on both sides. |
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| | :<math>((b+c)+(a+c)+(a+b))\left(\frac{1}{b+c}+\frac{1}{a+c}+\frac{1}{a+b}\right)\geq 9</math> |
|
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| </ul>
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| == GPS Ralph Lauren Online Outlet Australia ==
| | Which is true by the [[Cauchy-Schwarz inequality]]. |
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| | === Fifth proof === |
|
| | Starting from Nesbitt's inequality (1903) |
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| | :<math> \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq\frac{3}{2}</math>, |
|
| | we substitute a+b=x, b+c=y, c+a=z. |
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