Question

Sufficient conditions for Schur-concavity to imply concavity.

score 56 · Answer 1 · 2026-03-25 18:48:12

56

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Sufficient conditions for Schur-concavity to imply concavity.

Published on 25 Mar 2026 - 18:48

score 97 · Answer 2 · 2026-05-11 01:36:15

97

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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 11 May 2026 - 1:36

score 108 · Answer 3 · 2026-03-25 18:50:37

108

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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 25 Mar 2026 - 18:50

score 471 · Answer 4 · 2026-03-25 18:48:12

471

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

Published on 25 Mar 2026 - 18:48

score 267 · Answer 5 · 2026-03-25 18:49:40

267

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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 25 Mar 2026 - 18:49

score 120 · Answer 6 · 2026-03-25 18:49:45

120

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Trigonometric version of Schur's inequality

Published on 25 Mar 2026 - 18:49

score 2.7k · Answer 7 · 2026-03-25 18:48:06

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 25 Mar 2026 - 18:48

score 353 · Answer 8 · 2026-03-25 18:49:35

353

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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 25 Mar 2026 - 18:49

score 345 · Answer 9 · 2026-03-25 18:49:36

345

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

Published on 25 Mar 2026 - 18:49

score 157 · Answer 10 · 2026-03-25 18:49:37

157

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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 25 Mar 2026 - 18:49

score 191 · Answer 11 · 2026-03-25 18:50:17

191

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

Published on 25 Mar 2026 - 18:50

score 99 · Answer 12 · 2026-03-25 18:47:23

99

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Asking About Other Prove To Use In Solving This Inequality

Published on 25 Mar 2026 - 18:47

score 104 · Answer 13 · 2026-03-25 18:50:37

104

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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 25 Mar 2026 - 18:50

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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