Επώνυμες Ανισότητες

 Nesbitt's Inequality:

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

 Hölder's Inequality:

$(\sum _{i=1}^n{a}_i{b}_i\dots z_i)(\sum _{i=1}^n{x}_i{y}_i\dots w_i)\dots (\sum _{i=1}^n{m}_i{n}_i\dots k_i)\ge$ 

$\ge \left(\sum _{i=1}^n\sqrt[n]{a_i{x}_i\dots m_i}\right)\left(\sum _{i=1}^n\sqrt[n]{b_i{y}_i\dots n_i}\right)\dots {\left(\sum _{i=1}^n\sqrt[n]{z_i{w}_i\dots k_i}\right)}^n$

 Popoviciu's Inequality:

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

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

 Muirhead's Inequality:

$(\sum _{\text{sym}}{x}_1^{a_1}{x}_2^{a_2}\dots x_n^{a_n})\ge (\sum _{\text{sym}}{x}_1^{b_1}{x}_2^{b_2}\dots x_n^{b_n})$

 Vasc's Inequality:

$(a^2+b^2+c^2{)}^2\ge 3(a^{3}b+b^{3}c+c^{3}a)$

 Hardy-Littlewood-Pólya Inequality:

$\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{a}_i{a}_j{b}_i{b}_j\le \prod _{i=1}^n\left(\frac{a_i+b_i}{2}\right)$

 Vasile Cirtoaje's Inequality:

$\sum _{i=1}^n\sqrt{{\displaystyle \frac{a_i^2+b_i^2}{2}}}\ge \sqrt{\sum _{i=1}^n(a_i^2+b_i^2)}+\sqrt{\sum _{i=1}^n{a}_i{b}_i}$

 Mildorf's Inequality:

${\displaystyle \frac{a^3}{(b+c{)}^3}}+{\displaystyle \frac{b^3}{(c+a{)}^3}}+{\displaystyle \frac{c^3}{(a+b{)}^3}}\ge {\displaystyle \frac{3}{8}}$

 Inequality with Exponentials:

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

 APMO 1997 Inequality:

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

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

 Hardy's Inequality:

${\left({\displaystyle \frac{a_1+a_2+\dots +a_n}{n}}\right)}^{\frac{1}{a_1{a}_2\cdots a_n}}\le {\displaystyle \frac{a_1}{a_1}}+{\displaystyle \frac{a_2}{a_2}}+\dots +{\displaystyle \frac{a_n}{a_n}}$

 Pólya-Szegö Inequality:

${\displaystyle \frac{1}{2}}\sum _{i=1}^n\sum _{j=1}^n{a}_i{a}_j{b}_i{b}_j\le \prod _{i=1}^n\left({\displaystyle \frac{a_i+b_i}{2}}\right)$

 Vasile Cirtoaje's Inequality (Generalized):

$\sum _{i=1}^n\sqrt{{\displaystyle \frac{a_i^2+b_i^2}{2}}}\ge \sqrt{\sum _{i=1}^n(a_i^2+b_i^2)}+\sqrt{\sum _{i=1}^n{a}_i{b}_i}$

 Karamata's Inequality:

$\sum _{i=1}^n{a}_i{b}_{n+1-i}\ge \sum _{i=1}^n{a}_i{b}_i$

 Popoviciu's Inequality (Generalized):

$f(a)+f(b)+f(c)+3f\left({\displaystyle \frac{a+b+c}{3}}\right)\ge$ 

$\ge 2[f\left({\displaystyle \frac{a+b}{2}}\right)+f\left({\displaystyle \frac{b+c}{2}}\right)+f\left({\displaystyle \frac{c+a}{2}}\right)]$

 Rearrangement Inequality:

${\displaystyle \frac{a_1+a_2+\dots +a_n}{n}}\ge \sqrt[n]{a_1{a}_2\cdots a_n}$

 Inequality Bernoulli 

$(1+x{)}^n\ge 1+nx$, $x>-1$

$(1+x{)}^n\le 1+nx$, $o<x<1$