For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.

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2
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1answer
2k views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
25
votes
1answer
498 views

Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
3
votes
1answer
199 views

Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
3
votes
2answers
84 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...
10
votes
3answers
615 views

How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$

let $f$ on $[a,b]$ two continuously differentiable functions,such $$f(a)=f(b)=0, f'(a)=1,f'(b)=0,b>a>0$$ show that $$\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$$ My idea: use ...
14
votes
1answer
321 views

Inequality of numerical integration $\int _0^\infty x^{-x}\,dx$.

There was a friend asking me how to prove $$\int_0^\infty x^{-x}\,dx<2$$ Mathematica showed that its approximate value is 1.99546, so I think it isn't easy to solve it, can you provide me some ...
10
votes
6answers
879 views

How prove this $\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx$

let $f\in C^{(1)}[a,b]$,and such that $f(a)=f(b)=0$, show that $$\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx\cdots\cdots (1)$$ My try: use Cauchy-Schwarz inequality we have ...
12
votes
6answers
478 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
8
votes
1answer
352 views

Show inequality of integrals (cauchy-schwarz??)

$f:[0,1]\to\mathbb{C}$ continuous and differentiable and $f(0)=f(1)=0$. Show that $$ \left |\int_{0}^{1}f(x)dx \right |^2\leq\frac{1}{12}\int_{0}^{1} \left |f'(x)\right|^2dx $$ Well I know that $$ ...
6
votes
4answers
367 views

How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$

Let $f\in C^{1}[0,1]$ such that $f(0)=f(1)=0$. Show that $$\left(\int_{0}^{1}f(x)dx\right)^2\le\dfrac{1}{12}\int_{0}^{1}|f'(x)|^2dx.$$ I think we must use Cauchy-Schwarz inequality ...
5
votes
0answers
115 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
2
votes
1answer
1k views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
5
votes
1answer
457 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
5
votes
2answers
151 views

How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
3
votes
4answers
197 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...
2
votes
1answer
1k views

Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
8
votes
1answer
85 views

How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$

let $$f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$$ show that: $$e^x|f(x)|\le 2$$ My idea: let $$e^t=u$$ then $$|f(x)|=|\int_{e^x}^{e^{x+1}}\dfrac{1}{u}d\cos{u}|$$
29
votes
2answers
734 views

On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$ It's easy to ...
13
votes
4answers
4k views

Jensen's inequality for integrals

What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways. Supposing that $\varphi$ is a convex function on the real line and $g$ is ...
14
votes
2answers
572 views

Proof of bound on $\int t\,f(t)\ dt$ given well-behaved $f$

I got the following question by mail from someone I don't know from Adam. (Quoted in part.) if $f(t)$ continuously diff. on $[0,1]$ and a) $\int_0^1f(t)\ dt=0$ b) $m\le f\,'\le M$ on ...
8
votes
1answer
1k views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
8
votes
1answer
446 views

How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$

let $f(x)\ge 0,x\in [0,1]$, and is increasing in $[0,1]$ show that: There exists $g(x)\ge 0,x\in [0,1]$,and $g''(x)>0$, such $g(x)\le f(x)$, and such ...
6
votes
4answers
736 views

Proof of Schur's test via Young's inequality

I am able to prove the following generalization of Schur's test using the Riesz-Thorin interpolation theorem, however I have been stuck for days now trying to prove it using Young's inequality: Let ...
5
votes
2answers
468 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
4
votes
0answers
110 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
1
vote
2answers
350 views

Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$

Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$ My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...
0
votes
2answers
568 views

How to prove equality from poincare inequality?

Let $$D = \{y \in C^1(0,1) : y(0) = y(1) = 0\}$$ Suppose there exists a $C_0$ such that $$\int_{0}^{1} y^2 \ dx \leq C_0 \int_{0}^{1} (y')^2 \ dx$$ for all $y \in D$, and for all $C < C_0$ ...
3
votes
1answer
435 views

integral inequality with derivative

Let a function $f:[0,1]→\mathbb{R}$ have a continuous derivative and $$\int_{0}^{1}f(x)dx=0$$ Show that for every $\alpha \in [0,1]$, $$\left|\int_{0}^{\alpha}f(x)dx\right|\le\frac{1}{8} \mathbb{sup} ...
2
votes
1answer
30 views

Unique weak solution to Helmholtz equation on a square

I've recently started studying the modern theory of PDEs. I studied some basic properties of Sobolev space and then started with linear elliptic PDEs. I consider the following problem: For which ...
2
votes
1answer
48 views

Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $

If $f$ is real-valued and continuously differentiable on $\mathbb{R}$, prove that $$ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $$ Attempt: I tried the ...
2
votes
1answer
109 views

Upperbound for integral of a function times a cosine

Let $f$ be a function such that $0<c_1\leqslant f(x)\leqslant c_2$ for all $x$, and $g$ be a positive function. Assuming that we know the integral $\int_0^\infty g(x)\cos{x}\,\mathrm{d}x$, is it ...
1
vote
1answer
79 views

Find the upper bound of the derivative of an analytic function

The question is: if $f(z)$ is analytic and $|f(z)|\leq M$ for $|z|\leq r$, find an upper bound for$|f^{(n)}(z)|$ in $|z|\leq\frac{r}{2}$. My attempt: Since $f(z)$ is analytic where $|z|\leq r$, we ...
1
vote
2answers
33 views

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, proving $WZ_n\rightarrow 0$

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, I need to prove $WZ_n\rightarrow 0$. I was given a hint: show that for every $\delta,\epsilon<0$ we have ...
1
vote
1answer
81 views

use plancherel theorem to prove an integral inequality

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R). I'm sorry I don't know how to use Latex to post the problem. The origional problem is here: ...
1
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1answer
1k views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
0
votes
1answer
66 views

A generalisation of Cauchy–Schwarz

Let $u_1,\dots,u_n$ be vectors in an inner-product space and define $a_{ij} = \langle u_i , u_j \rangle$. So $$ A=(a_{ij}) = \begin{pmatrix} u_1^{\dagger} \\ \vdots\\ u_n^{\dagger} \end{pmatrix} ...
-3
votes
1answer
87 views

Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$