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

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21 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 ...
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1answer
41 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
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97 views
+50

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 ...
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0answers
18 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 $$ ...
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36 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 ...
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1answer
23 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)} ...
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2answers
56 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 ?
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3answers
121 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 ...
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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 ...
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24 views

Monoticity of power means.

An integral power mean of $f:\mathbb{R}\supset [a,b]\rightarrow \mathbb{R}$ is defined as \begin{equation} M_r(f,x)=\begin{cases} \Big\{\frac{1}{b-a}\int_a^b ...
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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 ...
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0answers
17 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 ...
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0answers
21 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 ...
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1answer
50 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 ...
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1answer
26 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 ...
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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 ...
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41 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 ...
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25 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 $f(x) \leq K + \left|\int_{x_0}^x f(t)g(t) \ \mathrm{d}t\right|,$ ...
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1answer
47 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 ...
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3answers
96 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 ...
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2answers
26 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 ...
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4answers
89 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 \ ...
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4answers
185 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 ...
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1answer
29 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:\; ...
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26 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 ...
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2answers
33 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?
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1answer
76 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 ...
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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 ...
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1answer
29 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 ...
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1answer
77 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 ...
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1answer
273 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 ...
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2answers
235 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 $$\int_a^b f(x) d x = f\left(\frac{a+b}{2}\right) = 0$$ ...
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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 ...
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2answers
102 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 ...
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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 $$ ...
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84 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 ...
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4answers
380 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): $$ \sqrt{\left(f(b)-f(a)\right)^2+\left(b-a\right)^2}\le\int_a^b \sqrt{1+f'(x)^2}dx $$
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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 ...
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3answers
198 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 ...
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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 ...
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42 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
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1answer
54 views

Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
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2answers
60 views

Questioning a Proof of Khinchine Inequality

[Khinchine Inequality] Let $a_1,\ldots,a_n\in R$, $\varepsilon_1,\ldots,\varepsilon_n$ be i.i.d. Rademacher random variables: $P(\varepsilon_i=1)=P(\varepsilon_i=-1)=0.5$, and $0<p<\infty$. Then ...
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3answers
216 views

An integral inequality with inverse

Let $f:[0,1]\to [0,1]$ be a non-decreasing concave function, such that $f(0)=0,f(1)=1$. Prove or disprove that : $$ \int_{0}^{1}(f(x)f^{-1}(x))^2\,\mathrm{d}x\ge \frac{1}{12}$$ A friend posed this to ...
7
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1answer
60 views

How to obtain the inequality $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p} $ from Jensen's inequality?

Let $f$ be a positive function with $\lVert f \rVert_{L^2}=1$. Let $p= 2n/(n-2)$. How to obtain $\int |f|^2\log(|f|) \leq \frac{n}{4}\log\lVert f \rVert^2_{L^p}$ from Jensen's inequality? Here all ...
6
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1answer
112 views

Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
3
votes
1answer
69 views

How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
7
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1answer
72 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 $$|f(x)|=|\int_{e^x}^{e^{x+1}}\dfrac{1}{u}d\cos{u}|$$
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0answers
27 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...