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

0
votes
1answer
28 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
1
vote
1answer
38 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
1
vote
2answers
67 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
3
votes
0answers
110 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
4
votes
0answers
209 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasi-linear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
1
vote
1answer
93 views

How to prove this integral-inequality.

Suppose $f$ is twice differentiable and satisfies $f(0)=0$. Prove the inequality. $$\int_0^1 |f(x)f'(x)| dx \le\ \frac{1}{2} \int_0^1 |f'(x)|^2 dx $$ This is a problem from undergraduate math ...
0
votes
0answers
22 views

Sufficient conditions for inequalities

Assume $f$ and $g$ are two Lebesgue integral functions on interval $[a,b]$. Do we have any nontrivial sufficient conditions on $f$ and $g$ such that $$\forall a\leq x<y\leq b,\quad ...
0
votes
1answer
25 views

Existance of solution of $Ax < b$

How to check if the inequality $Ax < b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$
1
vote
1answer
34 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
4
votes
0answers
23 views

Prove an integral inequality [closed]

It is true that $$ \int \int\int f(y) f(z) g(x-y)g(x-z)g(x) dx dy dz \leq || f||_2^2 ||g||_2^2 $$
0
votes
1answer
28 views

The best constant in an integral inequality

I find a interesting inequality. Suppose that $y=y(x)$ is a differentiable function in $(0,L)$ and $y(0)=y(a)=0$. Consider the fraction $$ F[y]=\frac{\int_0^{L}\vert y'\vert^2dx}{\int_0^L\vert ...
1
vote
1answer
49 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
4
votes
2answers
132 views

How to proof the following Gronwall type inequality?

Suppose that $g,k: [0,a] \to \mathbb R$ are continuous, $a >0 $, $\,k(t) \ge 0$,$\ c(t) \in C^1([0,a])$, $\, \dot c(t) \ge 0 $ (i.e. $c(t)$ is non decreasing) and $g(t)$ satisfies $$g(t) \le ...
1
vote
0answers
22 views

An integral inequality

Suppose $K\in C[0,1]^2$, $G\in C[0,1]$ are arbitrary and given. The question is that does there exists $H\in B[0,1]$ continuous a.e. with possibly finitely many discontinuities such that $$ ...
3
votes
0answers
50 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
1
vote
1answer
28 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
3
votes
2answers
64 views

An Inequality involving integration

Show that $$\int_{0}^{\pi} \left|\frac{\sin nx}{x}\right| dx \ge \frac{2}{\pi}\left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right)$$ How do I go about proving this inequality ?
5
votes
3answers
130 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 $f(x)\in C^{1}[0,1]$ ,and such $f(0)=f(1)=0$ 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 ...
0
votes
1answer
20 views

Show that $-2abX \le a^2Y +b^2Z\implies 4X^2 \le 4 Y Z $, $a,b \in \mathbb R $, $X;Y;Z \ge 0 $

Show that $-2abX \le a^2Y +b^2Z\implies 4X^2 \le 4YZ $ where $X,Y,Z $ are nonnegative and $a,b \in \mathbb R $. This looks almost as I could use Young's inequality, but not quiet. The above ...
2
votes
1answer
35 views

Integral inequality with sines

I am trying to show that there exists some constant $\mathcal{C}>0$ such that: $$\mathcal{C}\leq \int_0^1 |\sin (2\pi n x)-\sin (2\pi m x)|\;dx$$ For all distinct $m,n\in\mathbb{N}$. The constant ...
0
votes
0answers
26 views

Generalized Minkowski inequality for complex function

The Generalized Minkowski's inequality for any Borel function $f$ on $\mathbb{R}\times \mathbb{R}$ is $$ \int\Big(f(x,y)dx\Big)^2dy \le \Big[\int\Big(\int f^2(x,y)dy \Big)^{1/2}dx \Big]^2 $$ Does ...
1
vote
0answers
23 views

Inequality involving incomplete gamma function

When trying to answer this question: Find minimum $n$ such that $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$ has no answer inside the circle of radius $100$ centered at the origin I ended up in what ...
3
votes
1answer
54 views

$|f+g|^p$ Lebesgue-summable if $|f|^p$ and $|g|^p$ are

I read that the Minkowski integral inequality, which I knew for Riemann integrals on $[a,b]$, holds for Lebesgue integrals in the following form:$$\forall p\geq 1\quad\quad\Bigg(\int_X |f+g|^p ...
3
votes
1answer
28 views

How prove this integral inequality $\int_{0}^{\infty}(f(t))^2t^{-\delta}dt\le\frac{4}{(1-\delta)^2}\int_{0}^{\infty}(f'(t))^2t^{2-\delta}dt$?

Question: let $\delta\in(0,1)$, and $f\in C_{0}^{1}(R_{+})$,show that $$\int_{0}^{\infty}(f(t))^2t^{-\delta}dt\le\dfrac{4}{(1-\delta)^2}\int_{0}^{\infty}(f'(t))^2t^{2-\delta}dt$$ My idea: I ...
0
votes
0answers
29 views

Is there an inequality between the following quantities?

Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Is there an inequality between the following quantities? $$ \underset{|h|\leq ...
0
votes
0answers
46 views

Agmon and Young inequalities

I'm studying the book-Boundary control of PDEs using backstepping by M. Krstic and in Chapter 5, the authors makes a claim that I'm finding difficult to understand. We have the following two ...
1
vote
0answers
28 views

Related to Gronwall's Inequality.

The exercise is: Let $K \geq 0$, $f,g \geq 0$ continuous functions from $[a,b]$ to $\Bbb R$ and $x_0 \in ]a,b[$. Suppose that $f(x) \leq K + \left|\int_{x_0}^x f(t)g(t) \ \mathrm{d}t\right|,$ ...
3
votes
1answer
56 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
4
votes
3answers
99 views

Prove the inequality$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$

Prove: $$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$$ Here is my answer,but I want a different way to prove it. \begin{aligned} \int_0^{+\infty} {\sin x \over ...
0
votes
2answers
27 views

Inequalities with expected value on one side and probability on the other

In a part of a proof I am following, the author states that $$\displaystyle \mathbb{E}\left[\frac{|X_n - X|}{1 + |X_n - X|}\right] \leq \epsilon + \mathbb{P}(|X_n - X| > \epsilon)$$ and ...
4
votes
4answers
93 views

Prove that $2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$

I need some starting ideas, hints for proving that $$2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$$ I already checked that with Mathematica that numerically says that $$\int_0^1 \ ...
10
votes
4answers
189 views

Prove $1.43 < \displaystyle \int_0^1 e^{x^2} \mathrm{d}x < \frac{e+1}{2}$

Prove $$1.43<\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2}$$ What I did; As I have no idea how to approach the left inequality I work with $$\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2} \iff ...
0
votes
1answer
32 views

How can Use Gronwall for this PDE?

I'm trying to prove this. First I tried to multiply the equation by $\phi(x,t)$ and use the Gronwall Lemma, but it didn't work. Can anyone help? Here's the problem: Given a smooth field $u:\; ...
1
vote
0answers
28 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
1
vote
2answers
34 views

Integral mean inequalities

If $f \in C[0,1]$, then should be true that $$\left( \int |f|^p\right)^{1/p} \leq \left( \int |f|^q\right)^{1/q}$$ for $1<p \leq q$. However, I have found no sources on this fact. Is it true?
4
votes
1answer
79 views

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 ...
1
vote
1answer
64 views

$L_2$ error between a non-negative monotone function and its mean?

I have been recently trying to prove a lemma which seems true in every single example I have tried, yet that I didn't manage to prove so far unless making extra (not desirable) assumptions. A ...
0
votes
1answer
31 views

Integral Inequality involving the Euclidian Norm

I have spent several hours trying to establish the inequality shown in the attached photo. Here we assume that $\vec{r}(t)$ is a vector function in $R^n$, and is integrable on $[a,b]$. I am in need of ...
4
votes
1answer
81 views

An Integral Inequality

Let $f$ and $g$ be real functions such that $\int_0^\infty(f(x))^2dx<\infty$ and $\int_0^\infty(g(x)^2dx<\infty$. Prove that: $$\left(\int_0^\infty\int_0^\infty\frac{f(x)g(y)}{x+y}dxdy ...
3
votes
0answers
87 views
13
votes
1answer
276 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 ...
8
votes
2answers
236 views

A false integral inequality

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) d x = f\left(\frac{a+b}{2}\right) = 0$$ ...
1
vote
1answer
33 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
7
votes
2answers
103 views

Inequality of integrals $\int_0^1(f(x))^2 dx \geq 4$ if $\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$

If $$\int_0^1xf(x) dx=\int_0^1f(x) dx = 1$$ prove that $$\int_0^1(f(x))^2 dx \geq 4$$ EDIT My attempt is as follows - I can use only the $\int xf(x)$dx part to get a bound $\int f^2(x) dx \geq ...
1
vote
1answer
24 views

Show that $ \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $ \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$

Show that $\displaystyle \frac{2+\ln a }{2}\lt\frac{a-1}{\ln a} \lt \frac{1+a}{2}$ becomes $\displaystyle \frac{2(a-1)}{a+1}\lt\ln a \lt -1 + \sqrt{2a-1}$ The closest I can get is $$ ...
5
votes
0answers
91 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 ...
5
votes
4answers
383 views

Arc Length of a Curve

Let $f:[a,b]\to \mathbb{R}$ be a continuous function, how can you prove (not in the geometric way): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
0
votes
1answer
42 views

Integral inequality with gamma function

I have some trouble with paper I'm reading. The goal is this: let $s=\frac{1}{2}+\frac{1}{\log n}+it$. $M$ is a function such that $M(s)=O(\log^{3}(N(|t|+2)))$. Define $$U(s)=\frac{1}{2\pi ...
7
votes
3answers
204 views

Reverse Cauchy Schwarz for integrals

Let $f,g$ be two continuous positive functions over $[a,b]$ Let $m_1$ and $M_1$ be the minimum and maximum of $f$ Let $m_2$ and $M_2$ be the minimum and maximum of $g$ Prove that ...
1
vote
0answers
27 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...