Question

Prove the following refinement of AM-GM inequality

score 97 · Answer 1 · 2026-03-25 09:53:23

97

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Prove the following refinement of AM-GM inequality

Published on 25 Mar 2026 - 9:53

score 54 · Answer 2 · 2019-08-23 15:36:07

54

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Find the minimum value of this rational expression

Published on 23 Aug 2019 - 15:36

score 107 · Answer 3 · 2026-03-26 07:51:43

107

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If $a,b \in \mathbb{R}$ and $a+b\ge 0$, prove that $(a^2+b^2)^3\ge 32(a^3+b^3)(ab-a-b)$

Published on 26 Mar 2026 - 7:51

score 79 · Answer 4 · 2019-08-24 15:26:26

79

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show this inequaliy $\sum_{cyc}\frac{a}{|b-c|}\ge 3$

Published on 24 Aug 2019 - 15:26

score 138 · Answer 5 · 2026-03-27 07:16:55

138

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how to prove this inequality $\sum\frac{x}{(n-1)y+1}+\frac{1}{\sum x}\ge\frac{n+1}{n}$

Published on 27 Mar 2026 - 7:16

score 89 · Answer 6 · 2026-03-26 09:41:53

89

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Inequality $\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+(1-\frac{x+y}{2})^2\geq (1-x)^2+(1-y)^2$

Published on 26 Mar 2026 - 9:41

score 212 · Answer 7 · 2019-09-16 03:36:14

212

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Flaw in A.M.-G.M. inequality for finding the range of function.

Published on 16 Sep 2019 - 3:36

score 93 · Answer 8 · 2019-09-17 08:53:58

93

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If $a+b+c=2$ where $0 \leq {a,b,c} \leq 1$ find the range of $\frac{a}{1-a}\cdot\frac{b}{1-b}\cdot\frac{c}{1-c}$

Published on 17 Sep 2019 - 8:53

score 42 · Answer 9 · 2019-09-17 14:45:52

42

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when fix $a^2b+b^2a$,the minimal value of $a^2+ab+b^2$ occurs when $1\in\{a\}\bigcup\{b\}$

Published on 17 Sep 2019 - 14:45

score 338 · Answer 10 · 2019-09-18 08:36:19

338

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Prove $\frac{a}{b^{2}+1} + \frac{b}{c^{2}+1} + \frac{c}{a^{2} + 1} \ge \frac{3}{2}$

Published on 18 Sep 2019 - 8:36

score 971 · Answer 11 · 2019-09-18 19:17:53

971

Views

A sort of AM-GM inequality for matrices

Published on 18 Sep 2019 - 19:17

score 161 · Answer 12 · 2019-09-23 03:34:04

161

Views

Prove inequalities $\frac 34 \le I(a,b) \le 1$

Published on 23 Sep 2019 - 3:34

score 123 · Answer 13 · 2026-03-25 03:22:11

123

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Given three non-negative numbers $x,y,z$ so that $x+y+z=100$ and $x,y,z\notin (\!33,34\!)$. Prove that $xyz\leqq 33\cdot 34\cdot 33$ .

Published on 25 Mar 2026 - 3:22

score 194 · Answer 14 · 2019-09-23 14:39:00

194

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Prove that $\frac{\sqrt[n]{\prod_{k = 1}^nx_n}}{m} \ge n - 1$ where $\sum_{k = 1}^n\frac{1}{x_k + m} = \frac{1}{m}$.

Published on 23 Sep 2019 - 14:39

score 299 · Answer 15 · 2026-03-26 03:55:24

299

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What is the content of Young's inequality?

Published on 26 Mar 2026 - 3:55

List Question

Prove the following refinement of AM-GM inequality

Find the minimum value of this rational expression

If $a,b \in \mathbb{R}$ and $a+b\ge 0$, prove that $(a^2+b^2)^3\ge 32(a^3+b^3)(ab-a-b)$

show this inequaliy $\sum_{cyc}\frac{a}{|b-c|}\ge 3$

how to prove this inequality $\sum\frac{x}{(n-1)y+1}+\frac{1}{\sum x}\ge\frac{n+1}{n}$

Inequality $\left(1-\frac{x(1-x)+y(1-y)}{1-x+1-y}\right)^2+(1-\frac{x+y}{2})^2\geq (1-x)^2+(1-y)^2$

Flaw in A.M.-G.M. inequality for finding the range of function.

If $a+b+c=2$ where $0 \leq {a,b,c} \leq 1$ find the range of $\frac{a}{1-a}\cdot\frac{b}{1-b}\cdot\frac{c}{1-c}$

when fix $a^2b+b^2a$,the minimal value of $a^2+ab+b^2$ occurs when $1\in\{a\}\bigcup\{b\}$

Prove $\frac{a}{b^{2}+1} + \frac{b}{c^{2}+1} + \frac{c}{a^{2} + 1} \ge \frac{3}{2}$

A sort of AM-GM inequality for matrices

Prove inequalities $\frac 34 \le I(a,b) \le 1$

Given three non-negative numbers $x,y,z$ so that $x+y+z=100$ and $x,y,z\notin (\!33,34\!)$. Prove that $xyz\leqq 33\cdot 34\cdot 33$ .

Prove that $\frac{\sqrt[n]{\prod_{k = 1}^nx_n}}{m} \ge n - 1$ where $\sum_{k = 1}^n\frac{1}{x_k + m} = \frac{1}{m}$.

What is the content of Young's inequality?

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