Question

Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$:

score 57 · Answer 1 · 2026-03-23 13:45:39

57

Views

Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$:

Published on 23 Mar 2026 - 13:45

score 69 · Answer 2 · 2024-01-06 18:02:57

69

Views

Uniform bound for energy estimate

Published on 06 Jan 2024 - 18:02

score 130 · Answer 3 · 2024-01-08 20:05:09

130

Views

Prove the following inequality given a differential inequality.

Published on 08 Jan 2024 - 20:05

score 61 · Answer 4 · 2024-01-10 17:04:30

61

Views

$L^{p}$ norm of $\frac{|\nabla f|^{2}}{f}$

Published on 10 Jan 2024 - 17:04

score 52 · Answer 5 · 2024-01-11 06:14:38

52

Views

Limit representation of the Gamma function

Published on 11 Jan 2024 - 6:14

score 47 · Answer 6 · 2024-01-17 13:17:47

47

Views

Geometric-exponential series inequality

Published on 17 Jan 2024 - 13:17

score 116 · Answer 7 · 2024-01-22 17:14:42

116

Views

convex function inequality similr to the Hermite-Hadamard inequality

Published on 22 Jan 2024 - 17:14

score 65 · Answer 8 · 2024-02-05 09:27:30

65

Views

$L^p_x$ norms of $f(x, x+y)$

Published on 05 Feb 2024 - 9:27

score 64 · Answer 9 · 2024-02-16 05:35:21

64

Views

An integral inequality involving exponentials and $H^1(\mathbb{R}^2)$

Published on 16 Feb 2024 - 5:35

score 91 · Answer 10 · 2024-02-21 13:47:46

91

Views

Some integral inequality question with convexity

Published on 21 Feb 2024 - 13:47

score 76 · Answer 11 · 2024-02-25 16:56:06

76

Views

Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Published on 25 Feb 2024 - 16:56

score 25 · Answer 12 · 2024-03-01 16:20:47

25

Views

Theorem 7.20 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: The Comparison Theorem for Riemann-Stieltjes Integrals

Published on 01 Mar 2024 - 16:20

score 36 · Answer 13 · 2024-03-01 19:50:55

36

Views

Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$

Published on 01 Mar 2024 - 19:50

score 28 · Answer 14 · 2024-03-02 09:45:15

28

Views

Maximal mean value of a differentiable function with small variation

Published on 02 Mar 2024 - 9:45

score 204 · Answer 15 · 2024-03-02 16:06:27

204

Views

How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$?

Published on 02 Mar 2024 - 16:06

List Question

Find functions $f(x)$ and $g(x)$ such that the following conditions are satisfied for all $x > 0$:

Uniform bound for energy estimate

Prove the following inequality given a differential inequality.

$L^{p}$ norm of $\frac{|\nabla f|^{2}}{f}$

Limit representation of the Gamma function

Geometric-exponential series inequality

convex function inequality similr to the Hermite-Hadamard inequality

$L^p_x$ norms of $f(x, x+y)$

An integral inequality involving exponentials and $H^1(\mathbb{R}^2)$

Some integral inequality question with convexity

Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Theorem 7.20 in Apostol's MATHEMATICAL ANALYSIS, 2nd edition: The Comparison Theorem for Riemann-Stieltjes Integrals

Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$

Maximal mean value of a differentiable function with small variation

How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$?

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