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

learn more… | top users | synonyms

1
vote
1answer
40 views

A problem about center of mass

Suppose $f(x)$ is positive, increasing and Riemann-integrable on the interval $[a,b]$. Let$$\bar{x}=\frac{\int_{a}^{b}{xf(x)\text{d}x}}{\int_{a}^{b}{f(x)\text{d}x}}.$$Prove ...
2
votes
0answers
32 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
6
votes
2answers
104 views

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 ...
1
vote
0answers
26 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
0
votes
1answer
25 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
1
vote
2answers
37 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
0
votes
1answer
21 views

Somewhat L2 against H1 estimate; an inequality in H1

somehow I'm a little slow on this one: Let $\Omega = [0,1]^2 \subseteq \mathbb{R}^2$ and $\emptyset \neq D \subsetneq \Omega$. Do constants $c_1,c_2,c_2\in\mathbb{R}_{\geq 0}$ exist such that $$ c_1 ...
1
vote
0answers
35 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
1
vote
0answers
56 views

Integral inequality with a function twice differentiable

Let $f:[0,1]\longrightarrow\mathbb{R}$ be a function twice differentiable with continous second derivative and $f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ...
0
votes
0answers
15 views

Constructing a median-median inequality for a pair of distributions

Let $X_1$ and $X_2$ be continuous real random variables with medians $\mu_1$ and $\mu_2$ respectively. Can an effective upper bound on $|\mu_1-\mu_2|$ be found in terms of the variances $\sigma_{ij}$? ...
0
votes
1answer
23 views

An integral inequality with little information

$u,v$ are scalar fields on $V\subset\mathbb{R}^3$ such that $\nabla^2 u=0$ on $V$ and $u=v$ on $\partial V$. Prove that: $$\int_V|\boldsymbol{\nabla} ...
0
votes
0answers
19 views

Prove Inequality with Weight Function

The following question concerns a proof from Gilbarg and Trudinger's book titled "Elliptic Partial Differential Equations of Second Order", 2001, p.223 Without giving all the details of the proof in ...
7
votes
1answer
132 views

Integral Inequality $|\int_0^x f(t)dt|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|$

Let $f\in C^2(\mathbb{R})$ such that $f(1)=\int_0^1f(x)dx=0$. Prove that $$\left|\int_0^x f(t)dt\right|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|\, \forall x\in [0,1].$$ Thanks in advance!
1
vote
0answers
46 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
4
votes
1answer
82 views

Integral of composition [duplicate]

Prove that if $f,g:[0,1]\rightarrow[0,1]$ - continuous functions and f is strictly increasing then $$\int\limits_0^1f(g(x))dx\leq\int\limits_0^1f(x)dx+\int\limits_0^1g(x)dx.$$ I tried to prove that ...
3
votes
2answers
76 views

Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ ...
0
votes
1answer
51 views

How to apply Plancherel Theorem here?

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) and the L^1 norm of f(x) is less than or equal to 8 times (the L^2 norm of f(x)) times the L^2 norm ...
1
vote
1answer
37 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
vote
0answers
34 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
0
votes
1answer
68 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 ...
3
votes
4answers
130 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
26 views

A monotonicity property for ratios of power means

Let $Z$ be any non-degenerate positive random variable with pdf $% g(z)$. Let $a>0$ and $r\neq 0$ denote arbitrary real numbers. Define the "$r$ -mean" of $Z$ shifted by the constant $a$ as ...
-2
votes
1answer
66 views

Is the square of the integral of a real function less than the interval lenght times the integral of the square function?

$$\left(\int_0^T f(t)dt\right)^2 \leq T\cdot \int_0^T f(t)^2dt$$ Thank you.
0
votes
1answer
47 views

proof for inequality

Hi I am trying to understand a step of a proof, why is it true that $$\left [ \int_{-\infty}^{\infty}|t{f(t)}{f'^*(t)}|\,dt\right]^2\ge\left [ ...
2
votes
1answer
67 views

Cauchy-Schwarz-like inequality of integrals

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$ I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$ so if we ...
7
votes
2answers
137 views

How prove this inequality $ 1-\cos (xy) \le\int_0^xf(t) \sin {(tf(t))}dt + \int_0^y f^{-1}(t) \sin{(tf^{-1}(t))} dt .$

Question: Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: $$ 1-\cos (xy) \le\int_0^xf(t) ...
2
votes
1answer
45 views

Is $L^p \cap L^q$ dense in $L^r$?

It is known that $L^p \cap L^q \subset L^r$, where $1 \le p \le r \le q \le \infty$. Are all of these inclusions dense? I.e., do we have \begin{equation*} \overline{L^p \cap L^q} = L^r ...
1
vote
1answer
34 views

Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
2
votes
1answer
28 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
0
votes
1answer
30 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
1
vote
0answers
47 views

inequality question with integrals

There is a question that, I think, has a definite answer, but I can't figure it out. Given are three real valued functions, $f$,$g$, $w$, of a real variable $x$. The functions are non-negative, i.e., ...
0
votes
1answer
22 views

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 $$\dfrac{\partial u}{\partial x},\dfrac{\partial u}{\partial y},\dfrac{\partial^2 u}{\partial x\partial y}$$ ...
2
votes
2answers
100 views

Inequality for Expected Value of Product

Let $(\Omega, \mathbb{P}, \mathcal{F})$ be a probability space, and let $\mathbb{E}$ denote the expected value operator. Consider the random variables $f: \Omega \rightarrow \{0,1,2\}$ and $g: \Omega ...
11
votes
6answers
377 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, ...
3
votes
2answers
126 views

Integral inequality $\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$

How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
3
votes
1answer
137 views

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 ...
0
votes
2answers
151 views

Proving an inequality involving definite integrals.

Consider a probability distribution $f(x)$ with the assumption $\int_0^{\infty}f(x) > 1/2$. I am trying to prove the following inequality: \begin{align} \left( ...
1
vote
1answer
59 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
8
votes
1answer
129 views

A continuous function integral inequality

Let $m$ be a positive integer. $f\colon[0,\infty)\to[0,\infty)$ is a continuous function such that $f(f(x))=x^m,\forall x\in[0,\infty)$. Show that $$\int_0^1f^2(x)\,dx\ge\frac{2m-1}{m^2+6m-3}$$
0
votes
2answers
66 views

An Hardy-Littlewood like inequality

Let $\mu$ be a finite Borel measure on $R$. Show that $|\{x\in R : \sup_{r>0} \frac{1}{2r} \mu ([x-r,x+r]) \ge \lambda \}| \le \frac{C}{\lambda} \mu(R)$ for any $\lambda > 0$ and some absolute ...
3
votes
2answers
149 views

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: ...
6
votes
2answers
214 views

Prove that if f(x) is integrable, then so is e^(f(x)).

So here is my question: I'm working on a homework problem that deals with Jensen's Inequality. It is a rather simple application, I believe, but I'm a little stuck. Here is the problem, along with ...
1
vote
1answer
85 views

Maximum value in the Gruss inequality

The Gruss inequality states: $$\left|\frac{1}{x_2-x_1}\int_{x_1}^{x_2}dxf(x)g(x)-\frac{1}{(x_2-x_1)^2}\int_{x_1}^{x_2}dxf(x)\int_{x_1}^{x_2}dxg(x)\right|\le\frac{1}{4}(A-a)(B-b)$$ where $a\le f(x)\le ...
0
votes
1answer
120 views

How prove this integral inequality $\min_{x\in[0,1]}f(x)\ge-\int_{0}^{1}|f'(x)|dx$

let $f(x)$ can change sign in $x\in [0,1]$ and is continuous derivative function, show that $$\min_{x\in[0,1]}f(x)\ge-\int_{0}^{1}|f'(x)|dx$$ My try:since $f(x)$ change sign in $x\in [0,1]$, ...
4
votes
2answers
143 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 ...
0
votes
0answers
51 views

inequality of integrals

Let $f$ be of bounded variation on $[0,1]$ and $g:[0,1]\rightarrow \mathbb{R}$ be Lebesgue integrable on $[0,1]$. Prove \begin{equation} |\int_0^1fg d\lambda|\leq (|f(0)|+\text{Var}_{[0,1]}f)\cdot ...
0
votes
1answer
55 views

Upsetting Integral inequality

I have two smooth scalar non-negative functions f,g in $\mathbb{R}^+$ such that $$ f(x) \leq g(x), ~\forall x\in \mathbb{R}^+ $$ which are integrable with finite integrals in $\mathbb{R}^+$. I would ...
2
votes
1answer
75 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 ...
2
votes
1answer
23 views

An inequality in $L^p$-spaces

Let $\{f_k\}_{k=1}^{\infty}$ be a sequence in $L^p(\Omega,\Sigma,\mu)$ for $1\leq p<\infty$. Suppose $0<c=\inf_k \lVert f_k\rVert_p\leq \sup_k \lVert f_k\rVert_p=C<\infty$ and $f_if_j=0$ for ...
3
votes
2answers
57 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 ...