Question

Prove $\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4$ when $f(0)=f(1)=0$.

score 112 · Answer 1 · 2020-04-29 01:09:38

112

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Prove $\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4$ when $f(0)=f(1)=0$.

Published on 29 Apr 2020 - 1:09

score 2.6k · Answer 2 · 2020-04-29 06:01:16

2.6k

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Finding the maximum value of $\int_0^1 f^3(x)dx$

Published on 29 Apr 2020 - 6:01

score 222 · Answer 3 · 2020-04-30 19:18:28

222

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Proving that the solution to $f^{\prime}(x) = \frac{1}{x^{2} + (f(x))^{2}}$ is bounded above.

Published on 30 Apr 2020 - 19:18

score 414 · Answer 4 · 2020-05-02 03:19:20

414

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Given $\int_{\frac13}^{\frac23}f(x)dx=0$, how to prove $4860(\int_0^1f(x)dx)^2\le 11\int_0^1|f''(x)|^2dx$?

Published on 02 May 2020 - 3:19

score 95 · Answer 5 · 2020-05-03 12:28:02

95

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Prove or disprove that $\int_a^bf(x)g(x)dx\geqslant g(b)\frac{\int_a^bf(x)dx}{b-a}$.

Published on 03 May 2020 - 12:28

score 455 · Answer 6 · 2020-05-03 16:23:49

455

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Equality situation in Jensen's inequality

Published on 03 May 2020 - 16:23

score 74 · Answer 7 · 2020-05-06 20:26:27

74

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How to prove the following integral inequality?

Published on 06 May 2020 - 20:26

score 72 · Answer 8 · 2020-05-11 12:02:05

72

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Need help in proving a version of Stirling Formula

Published on 11 May 2020 - 12:02

score 128 · Answer 9 · 2026-02-23 03:28:34

128

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A question from Stein's book, Harmonic Analysis, oscillatory integral of the second kind.

Published on 23 Feb 2026 - 3:28

score 36 · Answer 10 · 2020-05-13 14:59:53

36

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Inequality Integral who involves logarithms and exponentials

Published on 13 May 2020 - 14:59

score 132 · Answer 11 · 2020-05-13 20:36:01

132

Views

Let $f: \mathbb R \rightarrow [0,\infty)$ be countinuously differentiable. Show that:

Published on 13 May 2020 - 20:36

score 278 · Answer 12 · 2020-05-16 14:50:25

278

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Wirtinger's inequality variation

Published on 16 May 2020 - 14:50

score 93 · Answer 13 · 2026-03-25 19:03:24

93

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References for variants of Jensen inequality

Published on 25 Mar 2026 - 19:03

score 73 · Answer 14 · 2020-05-21 07:15:10

73

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Reduction Formulae of $I_n=\int_0^\frac{\pi}{4}\tan^nx\, dx$ and $J_n=(-1)^nI_{2n}$

Published on 21 May 2020 - 7:15

score 224 · Answer 15 · 2020-05-22 12:54:08

224

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Show $\int_0^t (t-x)P_n(x)\,dx\leq \frac{t^2}{2}\int_0^1 P_n(x)\,\mathrm dx $ where $P_n(x)=(x(1-x))^{n}$

Published on 22 May 2020 - 12:54

List Question

Prove $\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4$ when $f(0)=f(1)=0$.

Finding the maximum value of $\int_0^1 f^3(x)dx$

Proving that the solution to $f^{\prime}(x) = \frac{1}{x^{2} + (f(x))^{2}}$ is bounded above.

Given $\int_{\frac13}^{\frac23}f(x)dx=0$, how to prove $4860(\int_0^1f(x)dx)^2\le 11\int_0^1|f''(x)|^2dx$?

Prove or disprove that $\int_a^bf(x)g(x)dx\geqslant g(b)\frac{\int_a^bf(x)dx}{b-a}$.

Equality situation in Jensen's inequality

How to prove the following integral inequality?

Need help in proving a version of Stirling Formula

A question from Stein's book, Harmonic Analysis, oscillatory integral of the second kind.

Inequality Integral who involves logarithms and exponentials

Let $f: \mathbb R \rightarrow [0,\infty)$ be countinuously differentiable. Show that:

Wirtinger's inequality variation

References for variants of Jensen inequality

Reduction Formulae of $I_n=\int_0^\frac{\pi}{4}\tan^nx\, dx$ and $J_n=(-1)^nI_{2n}$

Show $\int_0^t (t-x)P_n(x)\,dx\leq \frac{t^2}{2}\int_0^1 P_n(x)\,\mathrm dx $ where $P_n(x)=(x(1-x))^{n}$

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