For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
Questions tagged [integral-inequality]
1106 questions
7
votes
1 answer
hard integral inequality with $\pi$
a) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx$ converges
b) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx<1$
I have tried to find suitable integral sum for b), unsuccessfully. Is there a special method?
Booldy
- 1,070
7
votes
2 answers
Prove the following integral inequality: $\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2$
Let $f$ be a twice continuously differentiable function from $[0,1]$ into $\mathbb R$. Given that
$$f(0)+2f(\frac{1}{2})+f(1)=0,$$
show that
$$\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2.$$
I tried some methods, such as…
Johnson
- 134
6
votes
2 answers
How prove this $\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$
Show that this intergral inequality $$\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$$
I know this use Taylor's formula.But I think is very ugly,maybe this problem have simple methods.Thank you
math110
- 93,304
5
votes
3 answers
Determine whether the following integral inequality holds
Suppose that $\int_0^1 f dx = 0$. Is it true that
$$\left( \int_0^1 fg dx \right)^2 \leq \left( \int_0^1 g^2 dx - \left( \int_0^1 g dx \right)^2 \right) \left( \int_0^1 f^2 dx \right)$$
For context, this was stated as a true or false question on an…
Rellek
- 2,222
5
votes
2 answers
Poincaré inequality 1D
I am trying to prove the following result that I find in a book and which is denoted as Poincaré inequality.
Let $w$ be continuously differentiable on $(0,1)$. Then,
$\int_0^1 w^2(x)dx\leq 2 w^2(0)+4\int_0^1 w^2_x(x)dx$
where $w_x$ denotes the…
frank
- 279
5
votes
1 answer
Prove intergral inequality
If $f$ is a Riemann-integrable function on $[a,b]$ for which
$\int\limits_a^b f(x) dx = 0$, and $m \leq f(x) \leq M$ for all $a
\leq x \leq b$, then prove that $$\int\limits_a^b f(x)^2 dx \leq - m M
(b-a).$$
My only idea how to use…
Denis
- 91
4
votes
1 answer
Inequality with definite integrals
This problem has been bugging me for days. A function $f:[0,\,1]\to[0,\,1]$ with $f(0)=0$ and $f(1)=1$ is strictly increasing and differentiable, with $f'$ also strictly increasing. (So $f$ is a convex function.) I want to show that…
George Law
- 4,103
4
votes
2 answers
How prove this $\int_{a}^{b}xf(x)dx\le\int_{a}^{b}xg(x)dx$
let $f(x),g(x)$ is continuous on $[a,b]$,and such
$$\int_{a}^{x}f(t)dt\ge\int_{a}^{x}g(t)dt,x\in[a,b)$$
and
$$\int_{a}^{b}f(t)dt=\int_{a}^{b}g(t)dt$$
show that:
$$\int_{a}^{b}xf(x)dx\le\int_{a}^{b}xg(x)dx$$
my try: we only prove…
math110
- 93,304
4
votes
0 answers
How prove $\frac{\int_{0}^{1}xf^2(x)dx}{\int_{0}^{1}xf(x)dx}\le\frac{\int_{0}^{1}f^2(x)dx}{\int_{0}^{1}f(x)dx}$
let $f(x)$be positive and decreasing on $[0,1]$ show that:
$$\dfrac{\displaystyle\int_{0}^{1}xf^2(x)dx}{\displaystyle\int_{0}^{1}xf(x)dx}\le\dfrac{\displaystyle\int_{0}^{1}f^2(x)dx}{\displaystyle\int_{0}^{1}f(x)dx}$$
following is my…
math110
- 93,304
4
votes
0 answers
hard integral inequality with $e^{x^2}$
a) prove the convergency of
$$ \int_0^{\infty} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}$$
b) prove the inequality
$$ \int_0^{\frac {7\pi} {12}} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}<1$$
c) prove the inequality
$$ \int_0^{\frac {2\pi}…
Booldy
- 1,070
4
votes
4 answers
prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx < \frac{\pi^2}{64}$
prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx < \frac{\pi^2}{64}$
I showed that in
$$ \forall x \in [0,\frac{\pi}{4}] \quad \int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx \le \int_0^\frac{\pi}{4}…
Daniel Katzan
- 782
4
votes
3 answers
How prove this integral inequality $4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$
Question:
let $$f(0)=0,f(1)=1, f''(x)>0,x\in (0,1)$$
let $k>2$ are real numbers,show that
$$4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$$
This problem is from china Analysis problem book excise by (Min Hui XIE)
,analysis…
math110
- 93,304
3
votes
1 answer
Integrals on a closed ball
Prove
$$ \int_{\overline{B}(0,r)}c'D^{-1}c\ \exp(-1/2 y'D^{-1}y)\ dy >\int_{\overline{B}(0,r)}(y'D^{-1}c)^2\ \exp(-1/2 y'D^{-1}y)\ dy $$ where $ D\in\mathbb{R}^{n\times n} $ is a diagonal matrix, $ \overline{B}(0,r) $ is a closed $n$-ball with…
user41281
- 554
3
votes
1 answer
How prove this inequaliy $\dfrac{1}{b-a}\int_{a}^{b}|f(x)|dx>\dfrac{1}{2}|f(a)+f(b)|$
let $f(x)$ have twice differentiable on $[a,b]$,and such
$$f(x)\cdot f''(x)<0$$
show that
$$\dfrac{1}{b-a}\int_{a}^{b}|f(x)|dx>\dfrac{1}{2}|f(a)+f(b)|$$
I only know and can prove follow this…
math110
- 93,304
3
votes
2 answers
If $r\leq p\leq s$, prove that $\|f\|_p\leq \max(\|f\|_r,\|f\|_s)$
Let $X$ be a measure space with measure $\mu$ and $\|f\|_p=\left(\int_X |f|^p\; d\mu\right)^{\frac{1}{p}}$ be the standard $p$-norm for complex-valued $f$. I want to prove the following result, given $0
TerranDrop
- 840