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===Trivial Inequality=== | ===Trivial Inequality=== | ||
For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>. | For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>. | ||
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+ | Proof: We proceed by contradiction. Suppose there exists a real <math>x</math> such that <math>x^2<0</math>. We can have either <math>x=0</math>, <math>x>0</math>, or <math>x<0</math>. If <math>x=0</math>, then there is a clear contradiction, as <math>x^2 = 0^2 \not < 0</math>. If <math>x>0</math>, then <math>x^2 < 0</math> gives <math>x < \frac{0}{x} = 0</math> upon division by <math>x</math> (which is positive), so this case also leads to a contradiction. Finally, if <math>x<0</math>, then <math>x^2 < 0</math> gives <math>x > \frac{0}{x} = 0</math> upon division by <math>x</math> (which is negative), and yet again we have a contradiction. | ||
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+ | Therefore, <math>x^2 \ge 0</math> for all real <math>x</math>, as claimed. | ||
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===Arithmetic Mean/Geometric Mean Inequality=== | ===Arithmetic Mean/Geometric Mean Inequality=== | ||
For any set of real numbers <math>S</math>, <math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}</math> with equality iff <math>S_1=S_2=S_3...=S_{k-1}=S_k</math>. | For any set of real numbers <math>S</math>, <math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}</math> with equality iff <math>S_1=S_2=S_3...=S_{k-1}=S_k</math>. |
Latest revision as of 23:01, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 11. |
Inequalities
My favorite topic, saved for last.
Trivial Inequality
For any real , , with equality iff .
Proof: We proceed by contradiction. Suppose there exists a real such that . We can have either , , or . If , then there is a clear contradiction, as . If , then gives upon division by (which is positive), so this case also leads to a contradiction. Finally, if , then gives upon division by (which is negative), and yet again we have a contradiction.
Therefore, for all real , as claimed.
Arithmetic Mean/Geometric Mean Inequality
For any set of real numbers , with equality iff .
Cauchy-Schwarz Inequality
For any real numbers and , the following holds:
Cauchy-Schwarz Variation
For any real numbers and positive real numbers , the following holds:
.
Power Mean Inequality
Take a set of functions .
Note that does not exist. The geometric mean is . For non-negative real numbers , the following holds:
for reals .
, if is the quadratic mean, is the arithmetic mean, the geometric mean, and the harmonic mean.
RSM-AM-GM-HM Inequality
For any positive real numbers :
with equality iff .
Chebyshev's Inequality
Given real numbers and , we have
.
Minkowski's Inequality
Given real numbers and , the following holds:
Nesbitt's Inequality
For all positive real numbers , and , the following holds:
.
Schur's inequality
Given positive real numbers and real , the following holds:
.
Jensen's Inequality
For a convex function and real numbers and , the following holds:
Holder's Inequality
For positive real numbers , the following holds:
Muirhead's Inequality
For a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
Rearrangement Inequality
For any multi sets and , is maximized when is greater than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members of .
Newton's Inequality
For non-negative real numbers and the following holds:
,
with equality exactly iff all are equivalent.
MacLaurin's Inequality
For non-negative real numbers , and such that , for the following holds:
with equality iff all are equivalent.
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