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 43 · Answer 1 · 2026-04-15 22:06:06

43

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 15 Apr 2026 - 22:06

score 123 · Answer 2 · 2026-04-17 04:59:52

123

Views

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

Published on 17 Apr 2026 - 4:59

score 212 · Answer 3 · 2026-04-16 16:23:59

212

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 16 Apr 2026 - 16:23

score 285 · Answer 4 · 2026-04-13 17:28:46

285

Views

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

Published on 13 Apr 2026 - 17:28

score 129 · Answer 5 · 2026-04-13 22:19:22

129

Views

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

Published on 13 Apr 2026 - 22:19

score 88 · Answer 6 · 2026-04-15 23:35:19

88

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 15 Apr 2026 - 23:35

score 82 · Answer 7 · 2026-04-17 12:05:29

82

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 17 Apr 2026 - 12:05

score 29 · Answer 8 · 2026-04-15 23:35:18

29

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 15 Apr 2026 - 23:35

score 92 · Answer 9 · 2026-04-15 19:12:29

92

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 15 Apr 2026 - 19:12

score 255 · Answer 10 · 2026-04-17 16:02:01

255

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 17 Apr 2026 - 16:02

score 1k · Answer 11 · 2026-04-14 17:50:40

1k

Views

A problem from the Shortlist of the Romanian Mathematics Olympiad

Published on 14 Apr 2026 - 17:50

score 65 · Answer 12 · 2026-04-14 14:39:42

65

Views

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

Published on 14 Apr 2026 - 14:39

score 618 · Answer 13 · 2026-04-17 01:22:03

618

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 17 Apr 2026 - 1:22

score 348 · Answer 14 · 2026-04-13 13:05:12

348

Views

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

Published on 13 Apr 2026 - 13:05

score 47 · Answer 15 · 2026-04-11 21:17:05

47

Views

Unable to derive an inequality related to integrals

Published on 11 Apr 2026 - 21:17

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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