List questions from jensen-inequality

Published on 20 Jul 2020 - 18:47

82

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Is it true that $\mathbb{E}[X^n] > \mathbb{E}[X^{n-1}]\mathbb{E}[X]$ for all $n \geq 2$?

Published on 28 Jul 2020 - 19:40

128

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$(a+2)^3+(b+2)^3+(c+2)^3 \ge 81$ while $a+b+c=3$

Published on 20 Aug 2016 - 11:55

197

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Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

Published on 25 Mar 2026 - 11:13

72

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Is it inequality true

Published on 25 Mar 2026 - 4:35

210

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Prove that $\triangle ABC$ is equilateral, If $(a+b-c)^{a}\cdot (b+c-a)^{b}\cdot (c+a-b)^{c}\geq a^a\cdot b^b\cdot c^c$

Published on 25 Mar 2026 - 9:28

69

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Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$

Published on 19 Nov 2016 - 14:44

134

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Prove that $\prod\limits_{i=1}^nx_i^{x_i}\ge\prod\limits_{i=1}^n{x_i^a}$

Published on 14 Dec 2016 - 18:08

320

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Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$

Published on 17 Dec 2016 - 5:55

113

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Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$

Published on 23 Dec 2016 - 14:46

39

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The inequality $a^cb^d(c+d)^{c+d}\le c^cd^d(a+b)^{c+d}$

Published on 28 Dec 2016 - 12:45

225

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Prove that $x+y+z=3$ $\implies$ $x^xy^yz^z\ge1$

Published on 09 Jan 2017 - 23:23

283

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$\tan {\frac{A}{2}} + \tan {\frac{B}{2}} +\tan{\frac{C}{2}} \geq 4 - \sqrt {3} $

Published on 26 Jan 2017 - 9:32

151

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Inequalities with nice ideas

Published on 28 Jan 2017 - 13:13

153

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How to solve this problem using Jensen's inequality?

Published on 01 Feb 2017 - 17:20

List Question

Does Jensen's inequality still hold in general finite measure space?

Is it true that $\mathbb{E}[X^n] > \mathbb{E}[X^{n-1}]\mathbb{E}[X]$ for all $n \geq 2$?

$(a+2)^3+(b+2)^3+(c+2)^3 \ge 81$ while $a+b+c=3$

Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

Is it inequality true

Prove that $\triangle ABC$ is equilateral, If $(a+b-c)^{a}\cdot (b+c-a)^{b}\cdot (c+a-b)^{c}\geq a^a\cdot b^b\cdot c^c$

Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$

Prove that $\prod\limits_{i=1}^nx_i^{x_i}\ge\prod\limits_{i=1}^n{x_i^a}$

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$

Proving $\sqrt{x+y-x^2}+\sqrt{y+z-y^2}+\sqrt{z+w-z^2}+\sqrt{w+x-w^2}\le4\sqrt2-3$

The inequality $a^cb^d(c+d)^{c+d}\le c^cd^d(a+b)^{c+d}$

Prove that $x+y+z=3$ $\implies$ $x^xy^yz^z\ge1$

$\tan {\frac{A}{2}} + \tan {\frac{B}{2}} +\tan{\frac{C}{2}} \geq 4 - \sqrt {3} $

Inequalities with nice ideas

How to solve this problem using Jensen's inequality?

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