Επώνυμες Ανισότητες

 Nesbitt's Inequality:

$\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b} \geq \dfrac{3}{2}$

 Hölder's Inequality:

$\left( \sum_{i=1}^{n} a_i b_i \ldots z_i \right) \left( \sum_{i=1}^{n} x_i y_i \ldots w_i \right) \ldots \left( \sum_{i=1}^{n} m_i n_i \ldots k_i \right) \geq$ 

$\geq\left ( \sum_{i=1}^{n} \sqrt[n]{a_i x_i \ldots m_i} \right) \left( \sum_{i=1}^{n} \sqrt[n]{b_i y_i \ldots n_i} \right) \ldots \left( \sum_{i=1}^{n} \sqrt[n]{z_i w_i \ldots k_i} \right)^n$

 Popoviciu's Inequality:

$f(x) + f(y) + f(z) + 3f\left(\frac{x+y+z}{3}\right) \geq$ 

$2\left[f\left(\dfrac{x+y}{2}\right) + f\left(\dfrac{y+z}{2}\right) + f\left(\dfrac{z+x}{2}\right)\right]$

 Muirhead's Inequality:

$\left( \sum_{\text{sym}} x_1^{a_1} x_2^{a_2} \ldots x_n^{a_n} \right) \geq \left( \sum_{\text{sym}} x_1^{b_1} x_2^{b_2} \ldots x_n^{b_n} \right)$

 Vasc's Inequality:

$(a^2 + b^2 + c^2)^2 \geq 3(a^3 b + b^3 c + c^3 a)$

 Hardy-Littlewood-Pólya Inequality:

$\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} a_i a_j b_i b_j \leq \prod_{i=1}^{n} \left( \frac{a_i + b_i}{2} \right)$

 Vasile Cirtoaje's Inequality:

$\sum_{i=1}^{n} \sqrt{\dfrac{a_i^2 + b_i^2}{2}} \geq \sqrt{\sum_{i=1}^{n} (a_i^2 + b_i^2)} + \sqrt{\sum_{i=1}^{n} a_i b_i}$

 Mildorf's Inequality:

$\dfrac{a^3}{(b+c)^3} + \dfrac{b^3}{(c+a)^3} + \dfrac{c^3}{(a+b)^3} \geq \dfrac{3}{8}$

 Inequality with Exponentials:

$a^a b^b c^c \geq (abc)^{\frac{a+b+c}{3}}$

 APMO 1997 Inequality:

$a^3 + b^3 + c^3 + 2abc \geq ab \sqrt{2a^2 + 2b^2} +$

$+bc \sqrt{2b^2 + 2c^2} + ca \sqrt{2c^2 + 2a^2}$

 Hardy's Inequality:

$\left( \dfrac{a_1 + a_2 + \ldots + a_n}{n} \right)^{\frac{1}{a_1 a_2 \cdots a_n}} \leq \dfrac{a_1}{a_1} + \dfrac{a_2}{a_2} + \ldots + \dfrac{a_n}{a_n}$

 Pólya-Szegö Inequality:

$\dfrac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} a_i a_j b_i b_j \leq \prod_{i=1}^{n} \left( \dfrac{a_i + b_i}{2} \right)$

 Vasile Cirtoaje's Inequality (Generalized):

$\sum_{i=1}^{n} \sqrt{\dfrac{a_i^2 + b_i^2}{2}} \geq \sqrt{\sum_{i=1}^{n} (a_i^2 + b_i^2)} + \sqrt{\sum_{i=1}^{n} a_i b_i}$

 Karamata's Inequality:

$\sum_{i=1}^{n} a_i b_{n+1-i} \geq \sum_{i=1}^{n} a_i b_i$

 Popoviciu's Inequality (Generalized):

$f(a) + f(b) + f(c) + 3f\left(\dfrac{a+b+c}{3}\right) \geq$ 

$\geq2\left[f\left(\dfrac{a+b}{2}\right) + f\left(\dfrac{b+c}{2}\right) + f\left(\dfrac{c+a}{2}\right)\right]$

 Rearrangement Inequality:

$\dfrac{a_1 + a_2 + \ldots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$

 Inequality Bernoulli 

$(1+x)^n≥1+nx$, $x>-1$

$(1+x)^n≤1+nx$, $o<x<1$