Question

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

score 606 · Answer 1 · 2026-03-24 09:12:10

606

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 24 Mar 2026 - 9:12

score 358 · Answer 2 · 2026-03-22 23:02:56

358

Views

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

Published on 22 Mar 2026 - 23:02

score 143 · Answer 3 · 2026-03-20 00:12:56

143

Views

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 20 Mar 2026 - 0:12

score 370 · Answer 4 · 2026-03-22 17:53:51

370

Views

Interpretation of Jensen's inequality for the multivariate case

Published on 22 Mar 2026 - 17:53

score 486 · Answer 5 · 2026-03-22 23:02:20

486

Views

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

Published on 22 Mar 2026 - 23:02

score 78 · Answer 6 · 2026-03-23 11:31:29

78

Views

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

Published on 23 Mar 2026 - 11:31

score 310 · Answer 7 · 2026-03-23 05:21:17

310

Views

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

Published on 23 Mar 2026 - 5:21

score 121 · Answer 8 · 2026-03-24 03:47:12

121

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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 24 Mar 2026 - 3:47

score 109 · Answer 9 · 2026-03-24 10:30:23

109

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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 24 Mar 2026 - 10:30

score 105 · Answer 10 · 2026-03-21 10:42:38

105

Views

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

Published on 21 Mar 2026 - 10:42

score 107 · Answer 11 · 2026-03-23 08:21:54

107

Views

The existence of linear function which smaller than convex functions.

Published on 23 Mar 2026 - 8:21

score 166 · Answer 12 · 2026-03-22 05:28:54

166

Views

Jensen Inequality, a question about the proof and the statement

Published on 22 Mar 2026 - 5:28

score 152 · Answer 13 · 2026-03-22 14:45:26

152

Views

A simple application of Jensen's inequality

Published on 22 Mar 2026 - 14:45

score 86 · Answer 14 · 2026-03-21 18:58:50

86

Views

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

Published on 21 Mar 2026 - 18:58

score 457 · Answer 15 · 2026-03-21 03:01:10

457

Views

Inequality proof using Cauchy-Schwarz and Jensen's inequality

Published on 21 Mar 2026 - 3:01

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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