List questions from muirhead-inequality

Published on 08 Nov 2013 - 4:51

484

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The isoperimetric inequality for triangles

Published on 17 Jun 2017 - 3:58

54

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$a_1, ... . a_{n+1} > 0$, then $(\prod_{i=1}^{n+1}a_i)(\sum_{i=1}^{n+1} 1/a_i^n) \ge \sum_{i=1}^{n+1}a_i$?

Published on 10 Jul 2017 - 17:01

531

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Proving $a^{-2}bcd+b^{-2}cda+c^{-2}dab+d^{-2}abc\geqslant a+b+c+d$

Published on 12 Jul 2017 - 17:06

133

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Wild inequality

Published on 25 Mar 2026 - 6:04

165

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Prove the inequality $\sum_{cyc}\frac{a}{1+\left(b+c\right)^2}\le \frac{3\left(a^2+b^2+c^2\right)}{a^2+b^2+c^2+12abc}$

Published on 10 Aug 2017 - 15:19

112

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is there a generalization of Muirhead theorem for negatives reals?

Published on 15 Aug 2017 - 4:29

108

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Inequality : $\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}$

Published on 12 Sep 2017 - 13:18

90

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An inequality in four variables

Published on 22 Sep 2017 - 14:20

451

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$(xy+yz+zx)(1+6xyz) \geq 11xyz$

Published on 25 Sep 2017 - 21:24

514

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If $ab+bc+ca+abc=4$, then $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c$

Published on 07 Oct 2017 - 18:58

117

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from cyclic to symetric sums

Published on 02 Jan 2018 - 22:23

71

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Find the largest real $k$ such that: $(a+b+c)^2(ab+bc+ca)\geq k(a^2b^2+b^2c^2+c^2a^2)$

Published on 09 Jan 2018 - 10:07

191

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$\frac{(b+c-a)^2}{(b+c)^2+a^2}+ \frac{(c+a-b)^2}{(c+a)^2+b^2}+ \frac{(a+b-c)^2}{(a+b)^2+c^2} \ge \frac{3}{5}$

Published on 25 Mar 2026 - 13:56

147

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Proving $\sum_{cyc}\sqrt{a^4+a^2b^2+b^4}\geq \sum_{cyc} a\sqrt{2a^2+bc}$ for non-negative $a$, $b$, $c$

Published on 08 Aug 2020 - 2:01

List Question

How prove this inequality $\frac{x}{x+yz}+\frac{y}{y+zx}+\frac{z}{z+xy}\ge \frac{3}{2}$

The isoperimetric inequality for triangles

$a_1, ... . a_{n+1} > 0$, then $(\prod_{i=1}^{n+1}a_i)(\sum_{i=1}^{n+1} 1/a_i^n) \ge \sum_{i=1}^{n+1}a_i$?

Proving $a^{-2}bcd+b^{-2}cda+c^{-2}dab+d^{-2}abc\geqslant a+b+c+d$

Wild inequality

Prove the inequality $\sum_{cyc}\frac{a}{1+\left(b+c\right)^2}\le \frac{3\left(a^2+b^2+c^2\right)}{a^2+b^2+c^2+12abc}$

is there a generalization of Muirhead theorem for negatives reals?

Inequality : $\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}$

An inequality in four variables

$(xy+yz+zx)(1+6xyz) \geq 11xyz$

If $ab+bc+ca+abc=4$, then $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\leq 3\leq a+b+c$

from cyclic to symetric sums

Find the largest real $k$ such that: $(a+b+c)^2(ab+bc+ca)\geq k(a^2b^2+b^2c^2+c^2a^2)$

$\frac{(b+c-a)^2}{(b+c)^2+a^2}+ \frac{(c+a-b)^2}{(c+a)^2+b^2}+ \frac{(a+b-c)^2}{(a+b)^2+c^2} \ge \frac{3}{5}$

Proving $\sum_{cyc}\sqrt{a^4+a^2b^2+b^4}\geq \sum_{cyc} a\sqrt{2a^2+bc}$ for non-negative $a$, $b$, $c$

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