Question

Sufficient conditions for Schur-concavity to imply concavity.

score 55 · Answer 1 · 2023-04-18 08:23:10

55

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Sufficient conditions for Schur-concavity to imply concavity.

Published on 18 Apr 2023 - 8:23

score 95 · Answer 2 · 2023-08-07 04:09:46

95

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Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

Published on 07 Aug 2023 - 4:09

score 107 · Answer 3 · 2023-08-09 15:58:38

107

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If $x,y,z>0: x+y+z=3,$ prove $\frac{x^2}{y^2+yz+z^2}+\frac{y^2}{x^2+xz+z^2}+\frac{z^2}{y^2+yx+x^2}\ge \frac{5}{3}(x^2+y^2+z^2)+2xyz-6.$

Published on 09 Aug 2023 - 15:58

score 470 · Answer 4 · 2023-11-17 05:13:38

470

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How to prove $ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6$?

Published on 17 Nov 2023 - 5:13

score 266 · Answer 5 · 2023-11-23 10:30:46

266

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How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Published on 23 Nov 2023 - 10:30

score 119 · Answer 6 · 2023-12-24 16:44:50

119

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Trigonometric version of Schur's inequality

Published on 24 Dec 2023 - 16:44

score 2.7k · Answer 7 · 2013-09-30 15:42:35

2.7k

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How prove this $a+b+c+3\sqrt[3]{abc}\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})$

Published on 30 Sep 2013 - 15:42

score 352 · Answer 8 · 2021-08-22 13:16:07

352

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Prove the inequality $\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32$ where $a,b,c$ are positive reals.

Published on 22 Aug 2021 - 13:16

score 344 · Answer 9 · 2021-09-06 22:00:49

344

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If $x + y + z = 1$, what is $\min(xyz)$ where $x$, $y$ and $z$ are positive numbers?

Published on 06 Sep 2021 - 22:00

score 156 · Answer 10 · 2021-10-14 00:32:52

156

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In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Published on 14 Oct 2021 - 0:32

score 190 · Answer 11 · 2021-10-17 04:10:58

190

Views

Prove: $\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2\ge\sqrt{3}\sum_{cyc}{\sqrt[4]{\frac{5ab}{c}+4a}}$

Published on 17 Oct 2021 - 4:10

score 98 · Answer 12 · 2022-01-05 03:03:15

98

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Asking About Other Prove To Use In Solving This Inequality

Published on 05 Jan 2022 - 3:03

score 103 · Answer 13 · 2022-05-17 13:37:29

103

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Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

Published on 17 May 2022 - 13:37

List Question

Sufficient conditions for Schur-concavity to imply concavity.

Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

If $x,y,z>0: x+y+z=3,$ prove $\frac{x^2}{y^2+yz+z^2}+\frac{y^2}{x^2+xz+z^2}+\frac{z^2}{y^2+yx+x^2}\ge \frac{5}{3}(x^2+y^2+z^2)+2xyz-6.$

How to prove $ a+b+c+\sqrt{bc}+\sqrt{ca}+\sqrt{ab}\ge 6$?

How to prove $2\left(\sqrt{ab-1}+\sqrt{bc-1}+\sqrt{ca-1}\right)\le \left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\sqrt{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}.$?

Trigonometric version of Schur's inequality

How prove this $a+b+c+3\sqrt[3]{abc}\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ac})$

Prove the inequality $\sum_{cyc}\frac{a^2}{\sqrt{(b+c)(b^3+c^3)}}\geq\frac 32$ where $a,b,c$ are positive reals.

If $x + y + z = 1$, what is $\min(xyz)$ where $x$, $y$ and $z$ are positive numbers?

In triangle. Prove that: $2(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9+\sum_{cyc}{\sqrt{17+\frac{4(b+c)((b-c)^2-a^2)}{abc}}}$

Prove: $\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2\ge\sqrt{3}\sum_{cyc}{\sqrt[4]{\frac{5ab}{c}+4a}}$

Asking About Other Prove To Use In Solving This Inequality

Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

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