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
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$f(0)=0$,$f(1)=1$,find the largest $u$ such that $\int_0^1|f'(x)-f(x)|dx \geqslant u$.
Suppose
$E=\{f\in C^1[0,1]$: $f(0)=0$,$f(1)=1\}$.
Find the maximal $u\in \mathbb{R}$,such that forall $f\in E$,
\begin{align*}
\int_0^1|f'(x)-f(x)|dx \geqslant u.
\end{align*}
I can prove that $u\geqslant…
mbfkk
- 1,299
1
vote
1 answer
Integral Inequality 3 terms- Cauchy Schwarz
How can i demonstrate this inequality?
$(\int{f(x)g(x)h(x)dx})^2\le \int{f(x)g^2(x)dx}\int{f(x)h^2(x)dx}$
with,
$g, h$ arbitrary scalar functions and
$0 \le f(x)$
I tried to use cauchy inequality to 2 functions:
$(\int{f(x)g(x)h(x)dx})^2 \le…
user123
- 11
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0 answers
Integral Inequality Fubini Style
I'm struggeling to prove the following statement: Show that there exists a constant $C>0$ such that for all compactly supported, continuous and integrable functions $f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}_0^+$ there holds…
jim1970
- 91
1
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1 answer
Question on a proof of Wirtinger's Inequality
I've been looking for a proof of Wirtinger's Inequality that does not use Fourier Analysis (I have not learned Fourier Analysis yet). I found a proof online that I don't quite understand. I'll sketch out the details below.
Theorem: Let…
Hrhm
- 3,303
0
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1 answer
How to prove $\|u\|_\infty\le\|u\|_1+\|\nabla u\|_1+\|\partial^2_{xy}u\|_1$ on the square $[0,1]^2$
let $u(x,y)$ is continuous on $\Omega$,
$$\Omega=\{(x,y)|0\le x\le 1,0\le y\le 1\}$$
and
$$\dfrac{\partial u}{\partial x},\dfrac{\partial u}{\partial y},\dfrac{\partial^2 u}{\partial x\partial y}$$ are absolute integrable
show…
user94270
0
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1 answer
How prove this integral inequality.$\int_{1}^{e}f(a,b,c,x)dx\ge a+b+c$
show that
$$\int_{1}^{e}\dfrac{x^{a+b+c-1}[2(a+b+c)+(c+2a)x^{a-b}+(a+2b)x^{b-c}++(b+2c)x^{c-a}+(2a+b)x^{a-c}+(2b+c)x^{b-a}+(2c+a)x^{c-b}]}{(x^a+x^b)(x^b+x^c)(x^a+x^c)}dx\ge a+b+c$$
and This problem is from this…
math110
- 93,304
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1 answer
Integral inequality for positive functions
Is it true that $ \int_Sf(x)g(x)dx \leq \int_Sf(x)dx\int_Sg(x)dx \ \forall f(x),g(x)\geq0 $ ?
user742138
0
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1 answer
Show $\int_{0}^{1}(\int_{0}^{x}g(t)dt)^2 dx\leq\frac{1}{2}\int_0^1(1-x^2)(g(x))^2 dx$ for any $g(x)$ continuous
Prove (or disprove) that
$$\int_{0}^{1}\left(\int_0^x g(t)\ dt\right)^2dx\leq\frac{1}{2}\int_0^1 (1-x^2)(g(x))^2 dx$$
for any $g(x)$ continuous on $[0,1]$.
I have verified the cases of $g(x)$ being monomials like $x^k (k\in \mathbb{N})$ and found…
Y.Lin
- 157
0
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0 answers
How to prove integral inequality using Cauchy-Schwarz
Let $w >0$ and let $f: [−w, w]→R$ be a continuous function that is piecewise continuously differentiable on (−w, w). Assume that $f(x)≥0$ for all $x∈[−w, w]$ and $f(−w) =f(w) = 0$. Let a be the area under the graph of $f$ bounded above the x-axis…
User1825
- 5
0
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1 answer
Comparing an improper integral with the one for the same integrand multiplied by the integration variable
Consider the following improper integral, whereby $a \neq 0 $ is real and the $C^\infty$ real function $f(x)$ is further such that the following integral vanishes:
$$
\int_{-\infty} ^{\infty} \frac{f(x)}{a^2+x^2} \, dx \, = \, 0
$$
EDIT: I am now…
Luca
- 182
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1 answer
Why is this inequality true?
In a paper I'm reading, they say
Let $\delta$, $C_0$, and $n$ be positive constants. Then
$$\frac{\int_{\delta}^{\infty} \sqrt{n u^{\alpha}} e^{-nu^2 / (2C_0)}\,du}{\lambda_0^n} < \infty$$ for any fixed $\lambda_0 \in (e^{-\delta^2/(2C_0)},1)$.
I…
Kashif
- 1,497
0
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2 answers
Proof of Hardy's Classic Integral Inequality
While searching for different proofs of Hardy integral inequality, I saw a proof that used Homogeneity of norm and a kernel function. The first line of the proof says:
Let $$F(x)=\frac{1}{x}\int_{0}^{x}{f(t)dt}= \int_{0}^{1}{f(tx)dt}.$$
The function…
avg_ali
- 35
0
votes
1 answer
Integral inequality for sin function
Let $0
Sigur
- 6,416
- 3
- 25
- 45
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Integration Inequalities
The question we need to prove is
$$1.222\le 1+3^{-2}+5^{-2}+\cdots\le 1.252.$$
I know how to prove the first part
$$1+3^{-2}+...+43^{-2}\ge 1.222.$$
But I do not know how to prove the second part.
If I upper bound it using $1.222 + 45^{-2} +…
user614642
- 45
