Question

Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

score 186 · Answer 1 · 2019-04-15 13:40:58

186

Views

Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

Published on 15 Apr 2019 - 13:40

score 2.5k · Answer 2 · 2019-04-15 17:12:18

2.5k

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Is it always true that integral of nonnegative function is non negative

Published on 15 Apr 2019 - 17:12

score 102 · Answer 3 · 2019-04-16 14:11:28

102

Views

Proof-Verification: $ \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

Published on 16 Apr 2019 - 14:11

score 114 · Answer 4 · 2019-04-17 04:15:24

114

Views

$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

Published on 17 Apr 2019 - 4:15

score 223 · Answer 5 · 2019-04-19 16:27:40

223

Views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Published on 19 Apr 2019 - 16:27

score 59 · Answer 6 · 2019-04-19 21:28:25

59

Views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Published on 19 Apr 2019 - 21:28

score 27 · Answer 7 · 2019-04-20 00:08:33

27

Views

About the sufficient conditions for an upperbound for an integral ratio.

Published on 20 Apr 2019 - 0:08

score 28 · Answer 8 · 2019-04-21 15:17:17

28

Views

An integral inequality on a disc about twice derivatives.

Published on 21 Apr 2019 - 15:17

score 53 · Answer 9 · 2019-04-27 20:41:26

53

Views

Prove or disprove following inequality of integrals

Published on 27 Apr 2019 - 20:41

score 137 · Answer 10 · 2019-05-01 21:08:50

137

Views

The relation between the square of the integral and the integral of the square of the integrand

Published on 01 May 2019 - 21:08

score 43 · Answer 11 · 2019-05-03 12:29:28

43

Views

Minimizing the value of integral under certain conditions

Published on 03 May 2019 - 12:29

score 66 · Answer 12 · 2019-05-04 17:08:30

66

Views

Conditions to ensure an integral to be positive

Published on 04 May 2019 - 17:08

score 158 · Answer 13 · 2019-05-19 11:38:03

158

Views

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Published on 19 May 2019 - 11:38

score 66 · Answer 14 · 2019-05-25 07:58:08

66

Views

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

Published on 25 May 2019 - 7:58

score 81 · Answer 15 · 2019-06-12 03:54:34

81

Views

under which situation inequality between two integral yeilds pointwise inequality?

Published on 12 Jun 2019 - 3:54

List Question

Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

Is it always true that integral of nonnegative function is non negative

Proof-Verification: $ \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

About the sufficient conditions for an upperbound for an integral ratio.

An integral inequality on a disc about twice derivatives.

Prove or disprove following inequality of integrals

The relation between the square of the integral and the integral of the square of the integrand

Minimizing the value of integral under certain conditions

Conditions to ensure an integral to be positive

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

under which situation inequality between two integral yeilds pointwise inequality?

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