Question

An olympiad-like inequality $\frac{x^5}{6x^4+5y^4}+\frac{y^5}{6y^4+5z^4}+\frac{z^5}{6z^4+5x^4}\geq \frac{x+y+z}{11}$

score 157 · Answer 1 · 2020-01-11 17:07:27

157

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An olympiad-like inequality $\frac{x^5}{6x^4+5y^4}+\frac{y^5}{6y^4+5z^4}+\frac{z^5}{6z^4+5x^4}\geq \frac{x+y+z}{11}$

Published on 11 Jan 2020 - 17:07

score 56 · Answer 2 · 2020-01-12 14:15:08

56

Views

How to show that if $\mathbb{E}(X^4) < \infty$, then $\mathbb{E}(X^2) < \infty$?

Published on 12 Jan 2020 - 14:15

score 68 · Answer 3 · 2020-01-13 21:50:33

68

Views

If $X$ is integrable and $\phi$ is concave, how to show $\phi (X)$ is integrable

Published on 13 Jan 2020 - 21:50

score 356 · Answer 4 · 2020-02-02 12:03:47

356

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Jensen's inequality for proving $E\left[\frac1X\right]\ge \frac1{E[X]}$ where $P(X\leq 0) = 0$

Published on 02 Feb 2020 - 12:03

score 306 · Answer 5 · 2026-03-25 08:12:50

306

Views

An exponential equation over positive real numbers $4^x+14^x+3^x=11^x+10^x$

Published on 25 Mar 2026 - 8:12

score 98 · Answer 6 · 2020-02-09 04:50:45

98

Views

If $f'' \ge 0$, $\int_0^2 f(x)dx \ge 2f(1)$

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score 165 · Answer 7 · 2020-02-13 20:33:46

165

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score 287 · Answer 8 · 2026-03-25 12:49:51

287

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How to prove $\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac b{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}$ when $a,b,c>0$

Published on 25 Mar 2026 - 12:49

score 1.4k · Answer 9 · 2020-02-23 07:47:21

1.4k

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Published on 23 Feb 2020 - 7:47

score 552 · Answer 10 · 2020-02-24 20:53:55

552

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score 138 · Answer 11 · 2020-02-26 06:28:09

138

Views

Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$

Published on 26 Feb 2020 - 6:28

score 145 · Answer 12 · 2020-02-27 00:19:16

145

Views

How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$

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score 171 · Answer 13 · 2020-03-13 00:06:46

171

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score 180 · Answer 14 · 2020-03-13 15:27:30

180

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score 125 · Answer 15 · 2026-02-23 10:17:35

125

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Jensen's inequality and convex Lagrangian

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List Question

An olympiad-like inequality $\frac{x^5}{6x^4+5y^4}+\frac{y^5}{6y^4+5z^4}+\frac{z^5}{6z^4+5x^4}\geq \frac{x+y+z}{11}$

How to show that if $\mathbb{E}(X^4) < \infty$, then $\mathbb{E}(X^2) < \infty$?

If $X$ is integrable and $\phi$ is concave, how to show $\phi (X)$ is integrable

Jensen's inequality for proving $E\left[\frac1X\right]\ge \frac1{E[X]}$ where $P(X\leq 0) = 0$

An exponential equation over positive real numbers $4^x+14^x+3^x=11^x+10^x$

If $f'' \ge 0$, $\int_0^2 f(x)dx \ge 2f(1)$

Sum of indicators and application of Jensen's inequality

How to prove $\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac b{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}$ when $a,b,c>0$

Proof of Jensen's inequality for convexity

Jensen's inequality and LOTUS applied to entropy in probability

Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$

How to prove that $\prod_{i=1} ^ \infty x_i^{a_i} \leq \sum_{i=1} ^ \infty a_ix_i$

Find the maximum value of a sum of cosines given certain condition

Prove special case of Jensen's inequality THE OPPOSITE WAY

Jensen's inequality and convex Lagrangian

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