Question

Prove that $\int_E \log fd\mu \leqslant \mu(E) \, \log \left[\frac{1}{\mu(E)} \right]$ for strictly positive measure $\mu$

score 613 · Answer 1 · 2026-05-06 13:30:39

613

Views

Prove that $\int_E \log fd\mu \leqslant \mu(E) \, \log \left[\frac{1}{\mu(E)} \right]$ for strictly positive measure $\mu$

Published on 06 May 2026 - 13:30

score 363 · Answer 2 · 2026-05-02 18:21:22

363

Views

Is it sufficient to prove Jensen's Inequality holds for an example probability distribution to prove that a function is convex?

Published on 02 May 2026 - 18:21

score 152 · Answer 3 · 2026-05-07 10:12:47

152

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How to prove the following ineqiulity : $\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,s)f(s)}ds$

Published on 07 May 2026 - 10:12

score 376 · Answer 4 · 2026-04-29 12:50:50

376

Views

Interpretation of Jensen's inequality for the multivariate case

Published on 29 Apr 2026 - 12:50

score 491 · Answer 5 · 2026-04-29 12:50:44

491

Views

Proving that $\frac {1}{3x^2+1}+\frac {1}{3y^2+1}+\frac {1}{3z^2+1}\geq \frac {3}{16 } $

Published on 29 Apr 2026 - 12:50

score 83 · Answer 6 · 2026-04-29 12:50:49

83

Views

Proving inequality for all real $a$ and $b$.

Published on 29 Apr 2026 - 12:50

score 313 · Answer 7 · 2026-04-21 07:54:07

313

Views

Demonstrate that $(x+y)\ln \left(\frac{x+y}{2}\right) \leq x\ln x +y\ln y$

Published on 21 Apr 2026 - 7:54

score 128 · Answer 8 · 2026-04-29 12:50:55

128

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Prove the inequality $\left(1+\frac{1}{a_1(1+a_1)}\right)...\left(1+\frac{1}{a_k(1+a_k)}\right)\ge\left(1+\frac{1}{p(1+p)}\right)^k$

Published on 29 Apr 2026 - 12:50

score 113 · Answer 9 · 2026-04-29 12:50:37

113

Views

Prove $\frac{x^{n}}{x+y^3}+\frac{y^{n}}{y+x^3} \geqslant \frac{2^{4-n}}{5}$ for $x, y > 0$ with $x+y=1$

Published on 29 Apr 2026 - 12:50

score 113 · Answer 10 · 2026-04-29 12:16:07

113

Views

Let $\sum\frac{1}{a^3+1}=2$. Prove that $\sum\frac{1-a}{a^2-a+1}\ge 0$

Published on 29 Apr 2026 - 12:16

score 114 · Answer 11 · 2026-05-02 18:18:49

114

Views

The existence of linear function which smaller than convex functions.

Published on 02 May 2026 - 18:18

score 175 · Answer 12 · 2026-04-22 23:27:37

175

Views

Jensen Inequality, a question about the proof and the statement

Published on 22 Apr 2026 - 23:27

score 156 · Answer 13 · 2026-04-24 07:14:55

156

Views

A simple application of Jensen's inequality

Published on 24 Apr 2026 - 7:14

score 91 · Answer 14 · 2026-04-23 10:23:25

91

Views

When is the integrand in Jensen's formula actually integrable?

Published on 23 Apr 2026 - 10:23

score 462 · Answer 15 · 2026-04-23 17:44:25

462

Views

Inequality proof using Cauchy-Schwarz and Jensen's inequality

Published on 23 Apr 2026 - 17:44

List Question

Prove that $\int_E \log fd\mu \leqslant \mu(E) \, \log \left[\frac{1}{\mu(E)} \right]$ for strictly positive measure $\mu$

Is it sufficient to prove Jensen's Inequality holds for an example probability distribution to prove that a function is convex?

How to prove the following ineqiulity : $\exp\left(\int_0^t f(s)ds \right) \le 1+ \int_0^t e^{\max(1,s)f(s)}ds$

Interpretation of Jensen's inequality for the multivariate case

Proving that $\frac {1}{3x^2+1}+\frac {1}{3y^2+1}+\frac {1}{3z^2+1}\geq \frac {3}{16 } $

Proving inequality for all real $a$ and $b$.

Demonstrate that $(x+y)\ln \left(\frac{x+y}{2}\right) \leq x\ln x +y\ln y$

Prove the inequality $\left(1+\frac{1}{a_1(1+a_1)}\right)...\left(1+\frac{1}{a_k(1+a_k)}\right)\ge\left(1+\frac{1}{p(1+p)}\right)^k$

Prove $\frac{x^{n}}{x+y^3}+\frac{y^{n}}{y+x^3} \geqslant \frac{2^{4-n}}{5}$ for $x, y > 0$ with $x+y=1$

Let $\sum\frac{1}{a^3+1}=2$. Prove that $\sum\frac{1-a}{a^2-a+1}\ge 0$

The existence of linear function which smaller than convex functions.

Jensen Inequality, a question about the proof and the statement

A simple application of Jensen's inequality

When is the integrand in Jensen's formula actually integrable?

Inequality proof using Cauchy-Schwarz and Jensen's inequality

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