Question

Prove an inequality without using Jensen's inequality

score 122 · Answer 1 · 2026-04-12 03:12:10

122

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Prove an inequality without using Jensen's inequality

Published on 12 Apr 2026 - 3:12

score 95 · Answer 2 · 2026-05-10 15:25:29

95

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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 10 May 2026 - 15:25

score 558 · Answer 3 · 2026-04-17 06:03:21

558

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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 17 Apr 2026 - 6:03

score 92 · Answer 4 · 2026-04-16 10:30:51

92

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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 16 Apr 2026 - 10:30

score 245 · Answer 5 · 2026-04-16 06:21:36

245

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Prove this inequality via weighted Jensen inequality

Published on 16 Apr 2026 - 6:21

score 172 · Answer 6 · 2026-04-13 13:34:46

172

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Jensen Inequality Proof Strong If and Only If Version

Published on 13 Apr 2026 - 13:34

score 75 · Answer 7 · 2026-04-17 12:54:20

75

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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 17 Apr 2026 - 12:54

score 113 · Answer 8 · 2026-05-10 15:26:10

113

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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 10 May 2026 - 15:26

score 99 · Answer 9 · 2026-04-14 20:33:45

99

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

Published on 14 Apr 2026 - 20:33

score 82 · Answer 10 · 2026-04-16 08:52:15

82

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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 16 Apr 2026 - 8:52

score 289 · Answer 11 · 2026-05-10 17:02:40

289

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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 10 May 2026 - 17:02

score 127 · Answer 12 · 2026-04-13 15:15:30

127

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Concavity of the expected value in mean and variance

Published on 13 Apr 2026 - 15:15

score 603 · Answer 13 · 2026-05-10 15:24:38

603

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

Published on 10 May 2026 - 15:24

score 2.6k · Answer 14 · 2026-04-14 00:33:10

2.6k

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

Published on 14 Apr 2026 - 0:33

score 4.7k · Answer 15 · 2026-04-15 23:50:51

4.7k

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

Published on 15 Apr 2026 - 23:50

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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