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 160 · Answer 1 · 2026-05-10 16:13:27

160

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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 10 May 2026 - 16:13

score 57 · Answer 2 · 2026-03-25 16:08:06

57

Views

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

Published on 25 Mar 2026 - 16:08

score 69 · Answer 3 · 2026-03-25 16:12:36

69

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

Published on 25 Mar 2026 - 16:12

score 357 · Answer 4 · 2026-03-25 16:08:29

357

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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 25 Mar 2026 - 16:08

score 307 · Answer 5 · 2026-05-10 18:28:51

307

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

Published on 10 May 2026 - 18:28

score 99 · Answer 6 · 2026-03-25 16:08:30

99

Views

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

Published on 25 Mar 2026 - 16:08

score 166 · Answer 7 · 2026-03-25 16:08:17

166

Views

Sum of indicators and application of Jensen's inequality

Published on 25 Mar 2026 - 16:08

score 288 · Answer 8 · 2026-05-10 22:58:15

288

Views

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 10 May 2026 - 22:58

score 1.5k · Answer 9 · 2026-03-25 16:08:35

1.5k

Views

Proof of Jensen's inequality for convexity

Published on 25 Mar 2026 - 16:08

score 553 · Answer 10 · 2026-03-25 16:08:35

553

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Published on 25 Mar 2026 - 16:08

score 139 · Answer 11 · 2026-03-25 16:08:26

139

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 25 Mar 2026 - 16:08

score 146 · Answer 12 · 2026-03-25 16:08:30

146

Views

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

Published on 25 Mar 2026 - 16:08

score 172 · Answer 13 · 2026-03-25 16:08:19

172

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score 181 · Answer 14 · 2026-03-25 16:08:15

181

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Prove special case of Jensen's inequality THE OPPOSITE WAY

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score 126 · Answer 15 · 2026-04-09 15:23:45

126

Views

Jensen's inequality and convex Lagrangian

Published on 09 Apr 2026 - 15:23

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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