Question

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

score 115 · Answer 1 · 2026-04-17 02:42:09

115

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

Published on 17 Apr 2026 - 2:42

score 2.6k · Answer 2 · 2026-04-13 18:26:08

2.6k

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

Published on 13 Apr 2026 - 18:26

score 225 · Answer 3 · 2026-04-16 13:26:55

225

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

Published on 16 Apr 2026 - 13:26

score 417 · Answer 4 · 2026-04-16 14:49:54

417

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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 16 Apr 2026 - 14:49

score 98 · Answer 5 · 2026-04-16 08:32:49

98

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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 16 Apr 2026 - 8:32

score 458 · Answer 6 · 2026-04-14 11:48:06

458

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Equality situation in Jensen's inequality

Published on 14 Apr 2026 - 11:48

score 77 · Answer 7 · 2026-04-08 21:37:57

77

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How to prove the following integral inequality?

Published on 08 Apr 2026 - 21:37

score 75 · Answer 8 · 2026-04-09 12:09:25

75

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Need help in proving a version of Stirling Formula

Published on 09 Apr 2026 - 12:09

score 131 · Answer 9 · 2026-04-24 10:46:57

131

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

Published on 24 Apr 2026 - 10:46

score 39 · Answer 10 · 2026-04-15 05:13:55

39

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Inequality Integral who involves logarithms and exponentials

Published on 15 Apr 2026 - 5:13

score 135 · Answer 11 · 2026-04-16 08:53:54

135

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

Published on 16 Apr 2026 - 8:53

score 281 · Answer 12 · 2026-04-15 07:31:48

281

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Wirtinger's inequality variation

Published on 15 Apr 2026 - 7:31

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 76 · Answer 14 · 2026-04-15 18:22:20

76

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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 15 Apr 2026 - 18:22

score 227 · Answer 15 · 2026-04-17 15:12:32

227

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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 17 Apr 2026 - 15:12

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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