Question

Show that $\mathbb{E}[\sup_{0 \leq s \leq t} M^4_s] \leq \frac43 \mathbb{E}[M_t \sup_{0 \leq s \leq t} M^3_s] $

score 40 · Answer 1 · 2020-02-12 13:59:04

40

Views

Show that $\mathbb{E}[\sup_{0 \leq s \leq t} M^4_s] \leq \frac43 \mathbb{E}[M_t \sup_{0 \leq s \leq t} M^3_s] $

Published on 12 Feb 2020 - 13:59

score 120 · Answer 2 · 2020-02-16 14:41:13

120

Views

How prove this integral inequality $\int_{0}^{1}x \sqrt{1+\{f'(x)\}^2}dx\le\ \frac{1}{\sqrt{2}}$

Published on 16 Feb 2020 - 14:41

score 209 · Answer 3 · 2020-02-19 07:30:27

209

Views

Find function $f(x)$ that is continuous on $[0,2]$ satisfies $f(2) = 3$; $\int_0^2 [f'(x)]^2 dx = 4$ and $\int_0^2 x^2f(x) dx = \frac{1}{3}$,

Published on 19 Feb 2020 - 7:30

score 283 · Answer 4 · 2020-02-19 21:47:11

283

Views

Is there a more general version of Young's inequality for integrals?

Published on 19 Feb 2020 - 21:47

score 126 · Answer 5 · 2020-02-22 03:51:22

126

Views

Prove isoperimetric inequality for a closed curve parametrized by a smooth $2\pi$-peirodic complex valued function

Published on 22 Feb 2020 - 3:51

score 85 · Answer 6 · 2020-02-26 07:51:07

85

Views

Can we show that $\int\left|1_Bf-\int_Bf\:{\rm d}\lambda\right|^2\:{\rm d}\lambda\le c\int\left|f-\int f\:{\rm d}\lambda\right|^2\:{\rm d}\lambda$?

Published on 26 Feb 2020 - 7:51

score 79 · Answer 7 · 2020-03-02 09:28:16

79

Views

Can we show $\int_{B_i}\left|f-\frac1{λ(B_i)}\int_{B_i}f\:{\rm d}λ\right|^2\:{\rm d}λ\le\int\left|f-\frac1{λ(B)}\int f\:{\rm d}λ\right|^2\:{\rm d}λ$?

Published on 02 Mar 2020 - 9:28

score 26 · Answer 8 · 2020-03-02 19:52:24

26

Views

Can we show $\int_{B_j}λ({\rm d}y)\frac{\left|h_j(y)f(y)-\frac{λ(h_jf)}{λ(B_j)}\right|^2}{r(y)}\le\int_Bλ({\rm d}y)\frac{|f(y)-γλf|^2}{r(y)}$?

Published on 02 Mar 2020 - 19:52

score 89 · Answer 9 · 2020-03-08 13:11:23

89

Views

$\left(\int_{K^c} X d \mu \right) \left(\int_{K} X d \mu \right) \le \frac{\mu(\Omega)}{4}\int_{\Omega} X^2 d \mu $

Published on 08 Mar 2020 - 13:11

score 252 · Answer 10 · 2020-03-10 19:24:14

252

Views

Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$

Published on 10 Mar 2020 - 19:24

score 1k · Answer 11 · 2020-03-11 11:42:49

1k

Views

A problem from the Shortlist of the Romanian Mathematics Olympiad

Published on 11 Mar 2020 - 11:42

score 62 · Answer 12 · 2020-03-11 19:54:04

62

Views

Is $\frac{\phi(r)}{r}$ dominated by $\phi'(r)$?

Published on 11 Mar 2020 - 19:54

score 616 · Answer 13 · 2020-03-12 19:54:04

616

Views

Prove that $\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)^{\!2}$

Published on 12 Mar 2020 - 19:54

score 345 · Answer 14 · 2020-03-13 22:54:55

345

Views

Does the Cauchy-Schwarz integral inequality still hold for convergent improper integrals?

Published on 13 Mar 2020 - 22:54

score 44 · Answer 15 · 2020-03-14 16:20:16

44

Views

Unable to derive an inequality related to integrals

Published on 14 Mar 2020 - 16:20

List Question

Show that $\mathbb{E}[\sup_{0 \leq s \leq t} M^4_s] \leq \frac43 \mathbb{E}[M_t \sup_{0 \leq s \leq t} M^3_s] $

How prove this integral inequality $\int_{0}^{1}x \sqrt{1+\{f'(x)\}^2}dx\le\ \frac{1}{\sqrt{2}}$

Find function $f(x)$ that is continuous on $[0,2]$ satisfies $f(2) = 3$; $\int_0^2 [f'(x)]^2 dx = 4$ and $\int_0^2 x^2f(x) dx = \frac{1}{3}$,

Is there a more general version of Young's inequality for integrals?

Prove isoperimetric inequality for a closed curve parametrized by a smooth $2\pi$-peirodic complex valued function

Can we show that $\int\left|1_Bf-\int_Bf\:{\rm d}\lambda\right|^2\:{\rm d}\lambda\le c\int\left|f-\int f\:{\rm d}\lambda\right|^2\:{\rm d}\lambda$?

Can we show $\int_{B_i}\left|f-\frac1{λ(B_i)}\int_{B_i}f\:{\rm d}λ\right|^2\:{\rm d}λ\le\int\left|f-\frac1{λ(B)}\int f\:{\rm d}λ\right|^2\:{\rm d}λ$?

Can we show $\int_{B_j}λ({\rm d}y)\frac{\left|h_j(y)f(y)-\frac{λ(h_jf)}{λ(B_j)}\right|^2}{r(y)}\le\int_Bλ({\rm d}y)\frac{|f(y)-γλf|^2}{r(y)}$?

$\left(\int_{K^c} X d \mu \right) \left(\int_{K} X d \mu \right) \le \frac{\mu(\Omega)}{4}\int_{\Omega} X^2 d \mu $

Prove that $\int_0^1\sqrt{f^4(x)+(\int_0^1f(t)\, dt)^4}\, dx\le \sqrt{2}\int_0^1f^2(x)\,dx$

A problem from the Shortlist of the Romanian Mathematics Olympiad

Is $\frac{\phi(r)}{r}$ dominated by $\phi'(r)$?

Prove that $\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)^{\!2}$

Does the Cauchy-Schwarz integral inequality still hold for convergent improper integrals?

Unable to derive an inequality related to integrals

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