User:Temperal/The Problem Solver's Resource11

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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 $x$ , $x^2\ge 0$ , with equality iff $x=0$ .

Proof: We proceed by contradiction. Suppose there exists a real $x$ such that $x^2<0$ . We can have either $x=0$ , $x>0$ , or $x<0$ . If $x=0$ , then there is a clear contradiction, as $x^2 = 0^2 \not < 0$ . If $x>0$ , then $x^2 < 0$ gives $x < \frac{0}{x} = 0$ upon division by $x$ (which is positive), so this case also leads to a contradiction. Finally, if $x<0$ , then $x^2 < 0$ gives $x > \frac{0}{x} = 0$ upon division by $x$ (which is negative), and yet again we have a contradiction.

Therefore, $x^2 \ge 0$ for all real $x$ , as claimed.

Arithmetic Mean/Geometric Mean Inequality

For any set of real numbers $S$ , $\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}$ with equality iff $S_1=S_2=S_3...=S_{k-1}=S_k$ .

Cauchy-Schwarz Inequality

For any real numbers $a_1,a_2,...,a_n$ and $b_1,b_2,...,b_n$ , the following holds:

$\left(\sum a_i^2\right)\left(\sum b_i^2\right) \ge \left(\sum a_ib_i\right)^2$

Cauchy-Schwarz Variation

For any real numbers $a_1,a_2,...,a_n$ and positive real numbers $b_1,b_2,...,b_n$ , the following holds:

$\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}$ .

Power Mean Inequality

Take a set of functions $m_j(a) = \left({\frac{\sum a_i^j}{n}}\right)^{1/j}$ .

Note that $m_0$ does not exist. The geometric mean is $m_0 = \lim_{k \to 0} m_k$ . For non-negative real numbers $a_1,a_2,\ldots,a_n$ , the following holds:

$m_x \le m_y$ for reals $x<y$ .

, if $m_2$ is the quadratic mean, $m_1$ is the arithmetic mean, $m_0$ the geometric mean, and $m_{-1}$ the harmonic mean.

RSM-AM-GM-HM Inequality

For any positive real numbers $x_1,\ldots,x_n$ :

$\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\fr...$

with equality iff $x_1=x_2=\cdots=x_n$ .

Chebyshev's Inequality

Given real numbers $a_1 \ge a_2 \ge ... \ge a_n \ge 0$ and $b_1 \ge b_2 \ge ... \ge b_n$ , we have

${\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}$ .

Minkowski's Inequality

Given real numbers $a_1,a_2,...,a_n$ and $b_1,b_2,\ldots,b_n$ , the following holds:

$\sqrt{\sum a_i^2} + \sqrt{\sum b_i^2} \ge \sqrt{\sum (a_i+b_i)^2}$

Nesbitt's Inequality

For all positive real numbers $a$ , $b$ and $c$ , the following holds:

${\frac{a}{b+c}} + {\frac{b}{c+a}} + {\frac{c}{a+b}} \ge {\frac{3}{2}}$ .

Schur's inequality

Given positive real numbers $a,b,c$ and real $r$ , the following holds:

$a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b)\ge 0$ .

Jensen's Inequality

For a convex function $f(x)$ and real numbers $a_1,a_2,a_3,a_4\ldots,a_n$ and $x_1,x_2,x_3,x_4\ldots,x_n$ , the following holds:

$\sum_{i=1}^{n}a_i\cdot f(x_i)\ge f(\sum_{i=1}^{n}a_i\cdot x_i)$

Holder's Inequality

For positive real numbers $a_{i_{j}}, 1\le i\le m, 1\le j\le n$ , the following holds:

$\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}$

Muirhead's Inequality

For a sequence $A$ that majorizes a sequence $B$ , then given a set of positive integers $x_1,x_2,\ldots,x_n$ , the following holds:

$\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}$

Rearrangement Inequality

For any multi sets ${a_1,a_2,a_3\ldots,a_n}$ and ${b_1,b_2,b_3\ldots,b_n}$ , $a_1b_1+a_2b_2+\ldots+a_nb_n$ is maximized when $a_k$ is greater than or equal to exactly $i$ of the other members of $A$ , then $b_k$ is also greater than or equal to exactly $i$ of the other members of $B$ .