Question

Prove an inequality without using Jensen's inequality

score 120 · Answer 1 · 2019-08-29 15:54:35

120

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Prove an inequality without using Jensen's inequality

Published on 29 Aug 2019 - 15:54

score 94 · Answer 2 · 2026-03-25 04:41:46

94

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Inequality $3\sin^{-1}\left(\frac{\pi^{1/3}}{2^{2/3}}\right)\leq a+b+c$ with some conditions

Published on 25 Mar 2026 - 4:41

score 555 · Answer 3 · 2019-09-07 12:52:54

555

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Inequality : $\Big(\frac{x^n+1+(\frac{x+1}{2})^n}{x^{n-1}+1+(\frac{x+1}{2})^{n-1}}\Big)^n+\Big(\frac{x+1}{2}\Big)^n\leq x^n+1$

Published on 07 Sep 2019 - 12:52

score 89 · Answer 4 · 2019-09-12 13:14:16

89

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Olympiad Inequality $\frac{ab+bc+ca}{a+b+c}(\sum_{cyc}\frac{1}{2a+b})\leq 1$ with proof (long post)

Published on 12 Sep 2019 - 13:14

score 242 · Answer 5 · 2019-09-12 18:31:48

242

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Prove this inequality via weighted Jensen inequality

Published on 12 Sep 2019 - 18:31

score 169 · Answer 6 · 2019-09-20 20:55:34

169

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Jensen Inequality Proof Strong If and Only If Version

Published on 20 Sep 2019 - 20:55

score 72 · Answer 7 · 2019-09-21 13:01:00

72

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Inequality : $\sum_{cyc}\Big(\frac{x^4}{8x^3+5y^3}\Big)^3\geq \frac{x^3+y^3+z^3}{13^3}$

Published on 21 Sep 2019 - 13:01

score 112 · Answer 8 · 2026-03-25 03:04:33

112

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Prove that $\frac{n}{3} \le \sum_{i = 1}^n\frac{(n - y)^2}{y^2(x_i^2 + x_i + 1) - 3ny(x_i + 1) + 3n^2} \le \sum_{i = 1}^n\frac{1}{x_i^2 + x_i + 1}$.

Published on 25 Mar 2026 - 3:04

score 97 · Answer 9 · 2019-09-24 00:38:25

97

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Proving $\operatorname{Var}( E[ X \mid Z ] ) \le \operatorname{Var}( X )$

Published on 24 Sep 2019 - 0:38

score 79 · Answer 10 · 2019-09-26 09:42:30

79

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Given a natural numbers $n$. Prove that $\left(\frac{\sum_{k=1}^{n}x_{k}}{n}\right)^{\sum\limits_{k=1}^{n}x_{k}}\leqq\prod_{k=1}^{n}x_{k}^{x_{k}}$ .

Published on 26 Sep 2019 - 9:42

score 287 · Answer 11 · 2019-09-27 12:42:12

287

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Nice olympiad inequality :$\frac{xy^2}{4y^3+3}+\frac{yz^2}{4z^3+3}+\frac{zx^2}{4x^3+3}\leq \frac{3}{7}$

Published on 27 Sep 2019 - 12:42

score 124 · Answer 12 · 2019-09-30 01:00:50

124

Views

Concavity of the expected value in mean and variance

Published on 30 Sep 2019 - 1:00

score 600 · Answer 13 · 2026-03-25 03:04:10

600

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Prove that $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

Published on 25 Mar 2026 - 3:04

score 2.6k · Answer 14 · 2014-11-05 22:08:24

2.6k

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Can Cauchy Schwarz inequality be proven using Jensen's inequality?

Published on 05 Nov 2014 - 22:08

score 4.7k · Answer 15 · 2014-12-19 05:38:47

4.7k

Views

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$

Published on 19 Dec 2014 - 5:38

List Question

Prove an inequality without using Jensen's inequality

Inequality $3\sin^{-1}\left(\frac{\pi^{1/3}}{2^{2/3}}\right)\leq a+b+c$ with some conditions

Inequality : $\Big(\frac{x^n+1+(\frac{x+1}{2})^n}{x^{n-1}+1+(\frac{x+1}{2})^{n-1}}\Big)^n+\Big(\frac{x+1}{2}\Big)^n\leq x^n+1$

Olympiad Inequality $\frac{ab+bc+ca}{a+b+c}(\sum_{cyc}\frac{1}{2a+b})\leq 1$ with proof (long post)

Prove this inequality via weighted Jensen inequality

Jensen Inequality Proof Strong If and Only If Version

Inequality : $\sum_{cyc}\Big(\frac{x^4}{8x^3+5y^3}\Big)^3\geq \frac{x^3+y^3+z^3}{13^3}$

Prove that $\frac{n}{3} \le \sum_{i = 1}^n\frac{(n - y)^2}{y^2(x_i^2 + x_i + 1) - 3ny(x_i + 1) + 3n^2} \le \sum_{i = 1}^n\frac{1}{x_i^2 + x_i + 1}$.

Proving $\operatorname{Var}( E[ X \mid Z ] ) \le \operatorname{Var}( X )$

Given a natural numbers $n$. Prove that $\left(\frac{\sum_{k=1}^{n}x_{k}}{n}\right)^{\sum\limits_{k=1}^{n}x_{k}}\leqq\prod_{k=1}^{n}x_{k}^{x_{k}}$ .

Nice olympiad inequality :$\frac{xy^2}{4y^3+3}+\frac{yz^2}{4z^3+3}+\frac{zx^2}{4x^3+3}\leq \frac{3}{7}$

Concavity of the expected value in mean and variance

Prove that $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

Can Cauchy Schwarz inequality be proven using Jensen's inequality?

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$

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