| 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_1b_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 $ | |
| Holder Inequality | |
| Tchebytchev Inequality | |
| Minkowski Inequality | |
| Cauchy-schwarz Inequality for Integrals | |
| Holder Inequality for Integrals | |
| Minkowski Inequality for Integrals | |
