keywords: triangle, Cauchy-Schwartz, Holder, Tchebytchev, Minkowski
Inequalities
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| Inequalities | |
|---|---|
| Triangular Inequalities | |
| $ \vert a_1 \vert - \vert a_2 \vert \leqq \vert a_1 +a_2 \vert \leqq \vert a_1 \vert + \vert a_2 \vert $ | |
| $ \vert a_1 + a_2 + \cdots + a_n \vert \leqq \vert a_1 \vert + \vert a_2 \vert + \cdots + \vert a_n \vert $ | |
| Cauchy-schwarz Inequality | |
| $ \vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert ^2 \leqq \left ( \vert a_1 \vert ^2 + \vert a_2 \vert ^2 + \cdots + \vert a_n \vert ^2 \right ) \left ( \vert b_1 \vert ^2 + \vert b_2 \vert ^2 + \cdots + \vert b_n \vert ^2 \right ) $ | |
| $ \mbox{ The equality is valid if and only if } a_1/b_1 = a_2/b_2 = \cdots = a_n/b_n $ | |
| Inequalities Involving Arithmetic, Geometric and Harmonic | |
| $ \mbox{ if } A, \ G \mbox{ and } H \mbox{ are arithmatic, geometric and harmonic means of a positive numbers } a_1 , a_2 ,\cdots , a_n, \mbox{ then } $ | |
| $ H \leqq G \leqq A $ | |
| $ A = \frac{a_1 + a_2 + \cdots + a_n}{n} \qquad \qquad G = \sqrt[n]{a_1a_2 \cdots a_n} \qquad \qquad \frac{1}{H} = \frac{1}{n} \left ( {1 \over a_1} + {1 \over a_2 }+ \cdots +{1 \over a_n } \right ) $ | |
| $ \mbox{ the equality occures only if } a_1 = a_2 =\cdots = a_n. $ | |
| Holder Inequality | |
| $ \vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert \leqq \left ( \vert a_1 \vert ^p + \vert a_2 \vert ^p + \cdots + \vert a_n \vert ^p \right ) ^{1/p} \left ( \vert b_1 \vert ^q + \vert b_2 \vert ^q + \cdots + \vert b_n \vert ^q \right ) ^{1/q} $ | |
| $ {1 \over p} + {1 \over q} = 1 \qquad p > 1, \ q > 1. $ | |
| $ \mbox{ The equality occures only if } \vert a_1 \vert ^{p-1} / \vert b_1 \vert = \vert a_2 \vert ^{p-1} / \vert b_2 \vert = \cdots =\vert a_n \vert ^{p-1} / \vert b_n \vert . $ | |
| $ \text{for} \ p = q = 2,\ \text{the formula reduces to Cauchy-Shwartz Inequality.} $ | |
| Tchebytchev Inequality | |
| $ \mbox{ if } a_1 \geqq a_2 \geqq \cdots \geqq a_n \mbox{ and } b_1 \geqq b_2 \geqq \cdots \geqq b_n \mbox{ then } $ | |
| $ \left ( \frac{a_1 + a_1 + \cdots + a_n}{n} \right ) \left ( \frac{ b_1 + b_2 + \cdots +b_n}{n} \right ) \leqq \frac{a_1b_1+a_2b_2+\cdots+a_nb_n}{n} $ | |
| $ (a_1 + a_2 + \cdots +a_n)(b_1 + b_2 + \cdots +b_n) \leqq n(a_1b_1 + a_2b_2 + \cdots +a_nb_n) $ | |
| Minkowski Inequality | |
| $ \mbox{ if } a_1,a_2, \cdots , a_n, b_1,b_2, \cdots, b_n \mbox{ are all positive and } p > 1 \mbox{ then } $ | |
| $ \left \{ (a_1+b+1)^p + (a_2+b_2)^p+ \cdots + (a_n+b_n)^p \right \} ^{1/p} \leqq (a_1^p + a_2^p + \cdots + a_n^p)^{1/p} + (b_1^p+b_2^p+ \cdots+ b_n^p)^{1/p} $ | |
| $ \mbox{ the equality holds if and only if } a_1/b_1 = a_2 /b_2 = \cdots = a_n/b_n. $ | |
| Cauchy-schwarz Inequality for Integrals | |
| $ \left \vert \int_a^b f(x) g(x) d x \right \vert ^2 \leqq \left \{ \int_a^b \vert f(x) \vert ^2 d x \right \}\left \{ \int_a^b \vert g(x) \vert ^2 d x \right \} $ | |
| $ \mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad $ | |
| Holder Inequality for Integrals | |
| $ \int_a^b \vert f(x) g(x) \vert d x \leqq \left \{ \int_a^b \vert f(x) \vert ^p d x \right \} ^{1/p} \left \{ \int _a^b \vert g(x) \vert ^q d x \right \} ^{1/q} $ | |
| $ \mbox { where } \frac{1}{p} + \frac{1}{q} = 1,\ p>1,\ q> 1. $ | |
| $ \mbox{ if } p = q =2, \mbox{ this formula reduces to Cauchy-Schwartz inequality for intergrals } \quad $ | |
| $ \mbox{ Equality holds only if } \vert f(x) \vert ^{p-1} / \vert g(x) \vert \mbox { is constant. } $ | |
| Minkowski Inequality for Integrals | |
| $ \mbox{ if } p > 1 , \quad $ | |
| $ \left \{ \int_a^b \vert f(x) + g(x) \vert ^p d x \right \} ^{1/p} \leqq \left \{ \int_a^b \vert f(x) \vert ^ p d x \right \} ^{1/p} + \left \{ \int_a^b \vert g(x) \vert ^p \right \} ^{1/p} $ | |
| $ \mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad $ | |
