Inequalities for Contests

1) $|a \cdot b| \leq \frac{1}{2}(a^2 + b^2)$

2) $\frac{a}{b} + \frac{b}{a} \geq 2$, if $\frac{a}{b} > 0$
It is an equation $\frac{a}{b} + \frac{b}{a} = 2$ when $a = b$.

$\frac{a}{b} + \frac{b}{a} \leq -2$, if $\frac{a}{b} < 0$
It is an equation $\frac{a}{b} + \frac{b}{a} = -2$ when $a = -b$.

3) $(a_1 + a_2 + \dotsb + a_n)(\frac{1}{a_1} + \frac{1}{a_2} + \dotsb + \frac{1}{a_n}) \geq n^2\\ a_i > 0, i = 1, 2,..., n$

4) Triangle inequality

For every two numbers $a_1$ and $a_2$ we have:
$||a_1| - |a_2|| \leq |a_1 \pm a_2| \leq |a_1| + |a_2|$

For n numbers the triangle inequality looks like:

$|a_1 + a_2 + \dotsb + a_n| \leq |a_1| + |a_2| + \dotsb + |a_n|$

5) $2^n > 2n + 1$, $n \in N$ and $n \geq 3$

6) Bernoulli's inequality

$(a + 1)^r > r\cdot a + 1$ when $a > 0, r \in Q, r > 1$

7) $1 + \frac{1}{1} + \frac{1}{1\cdot2} + \frac{1}{1\cdot2\cdot3} + \dotsb + \frac{1}{1\cdot2 \dotsb n} < 3 \\n\in N$

8) $(1 + \frac{1}{n})^n < 3$
when $n \ge 1, n \in N$

9) Inequality between the arithmetic mean, geometric mean, harmonic mean and root mean square

$H \leq G \leq A \leq S$
where:

$A = \frac{a_1 + a_2 + \dotsb + a_n}{n}$ - arithmetic mean

$G = \sqrt[n]{a_1a_2\dotsb a_n}$ - geometric mean

$H = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}}$ - harmonic mean

$S = \sqrt{\frac{a_1^2 + a_2^2 + \dots + a_n^2}{n}}$ - root mean square

If $a_1 = a_2 = \dots = a_n$ then $H = G = A = S$

10) Cauchy's Inequality

$G = \sqrt[n]{a_1a_2\dots a_n} \leq \frac{a_1 + a_2 + \dots + a_n}{n} = A$

$a_1^{\lambda_1} \dot a_2^{\lambda_2} \dots a_n^{\lambda_n} \leq \lambda_1a_1 + \lambda_2a_2 + \dots + \lambda_na_n$

where $\lambda_1 + \lambda_2 + \dots + \lambda_n = 1$ and $a_i > 0, i = 1, 2, ... n$

11) Chebyshev's inequality

If $a_1 \leq a_2 \leq ... \leq a_n$ and $b_1 \leq b_2 \leq ... \leq b_n$ then:

$\frac{a_1 + a_2 + \dots + a_n}{n} . \frac{b_1 + b_2 + \dots + b_n}{n} \leq \frac{a_1b_1 + a_2b_2 + \dots + a_nb_n}{n}$

It is an equation when $a_1 = a_2 = ... = a_n$ and $b_1 = b_2 = ... = b_n$.

12) Holder's inequality

If $\frac{1}{p} + \frac{1}{q} = 1, p > 1, q > 1$, then:

$|a_1b_1 + a_2b_2 + \dots + a_nb_n| \leq (|a_1|^p + |a_2|^p + \dots + |a_n|^p)^\frac{1}{p}(|b_1|^q + |b_2|^q + \dots + |b_n|^q)^\frac{1}{q}$

It is an equation when $\frac{|a_1|^{p-1}}{|b_1|} = \frac{|a_2|^{p-1}}{|b_2|} = \dots = \frac{|a_n|^{p-1}}{|b_n|}$

13) Cauchy-Schwarz(Bunyakovsky) inequality

Cauchy-Schwarz(Bunyakovsky) inequality is obtained by Holder's inequality when p = q = 2

$a_1b_1 + a_2b_2 + \dots + a_nb_n \leq \sqrt{(a_1^2 + a_2^2 + \dots + a_n^2)}\sqrt{(b_1^2 + b_2^2 + \dots + b_n^2)}$

It is an equality when $\frac{a_1}{b_1} = \frac{a_2}{b_2} = \dots = \frac{a_n}{b_n}$

15) Minkowski's inequality

a_i > 0, b_i > 0, i = 1, 2,...,n

$\sqrt{(a_1 + b_1)^2 + \dots + (a_n + b_n)^2} \leq \sqrt{(a_1 + b_1)^2} + \sqrt{(a_n + b_n)^2}$

It is an equation when
$\frac{a_1}{b_1} = \frac{a_2}{b_2} = \dots = \frac{a_n}{b_n}$

Minkowski's general inequality - if p > 0 then:

$((a_1 + b_1)^p + (a_2 + b_2)^p + \dots + (a_n + b_n)^p)^{\frac{1}{p}} \leq (a_1^p + a_2^p + \dots + a_n^p)^{\frac{1}{p}} + (b_1^p + b_2^p + \dots + b_n^p)^{\frac{1}{p}}$

It is an equation when
$\frac{a_1}{b_1} = \frac{a_2}{b_2} = \dots = \frac{a_n}{b_n}$