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

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0
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1answer
29 views

Why is the integral of a square always larger than the square of an integral?

I learned in physics that $\langle x^2 \rangle - \langle x \rangle ^2 = \sigma_x^2 \ge 0$ and thus $\langle x^2 \rangle \ge \langle x \rangle ^2$. In the case of continuous distribution, it becomes ...
0
votes
0answers
44 views

hard integral inequality with $e^{x^2}$

a) prove the convergency of $$ \int_0^{\infty} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}$$ b) prove the inequality $$ \int_0^{\frac {7\pi} {12}} \dfrac {\cosh x \cdot \sin x} {x\cdot ...
7
votes
1answer
125 views

hard integral inequality with $\pi$

a) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx$ converges b) Prove that $\int_{0}^{\infty} \arcsin{\pi^{-x^2}} dx<1$ I have tried to find suitable integral sum for b), unsuccessfully. ...
2
votes
0answers
36 views

integral inequality involving $\sup|f'|$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous function differentiable on $(0,1)$ with property that there exists $a \in (0,1]$ such that $$\int_{0}^a f(x)dx=0$$ Prove that $$\left|\int_{0}^1 ...
3
votes
1answer
108 views

Prove a function containing integrals is positive

From plots I find $$ U(x)=\int_x^1 t^{b-1}(1-t)^b dt-(1-2x)(1-x)^{b-1}x^b +2x\int_0^x t^{b-1}(1-t)^{b-1}dt \geq 0 $$ for any $x \in [0,\tfrac12], 0<b<1$, (or rather the plots of $\,\, ...
-2
votes
1answer
71 views

Inequalities help! [closed]

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

Integral inequality

Let $f:[0,1]\longrightarrow \mathbb{R}$ be a continuous function such that $$\int_{0}^{1}f(x)dx=0.$$ Is the following inequality true? $$2\left(\int_{0}^{1}xf(x)dx\right)^2\leq ...
4
votes
4answers
104 views

prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx < \frac{\pi^2}{64}$

prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx < \frac{\pi^2}{64}$ I showed that in $$ \forall x \in [0,\frac{\pi}{4}] \quad \int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx \le ...
8
votes
2answers
118 views

Two inequalities for $\left(\int_{a}^{b}|f(x)|^rdx\right)^{\frac{1}{r}}$

Show that if $f\in \mathcal C^{n+1}([a,b])$ and $f(a)=f^{'}(a)=\cdots=f^\left(n\right)(a)=0,$ then the following statements are ture: $\mathbf a)$ $ \forall r\in[1,\infty),$the inequality ...
9
votes
1answer
92 views

How prove this inequality $\left|\int_{a}^{b}f(x)dx\right|\leq \frac{1}{8}(b-a)^{2}\int_{a}^{b}\left|f''(x)\right|dx$

Let $f(x)\in \mathcal C^2([a,b]),f(\frac{a+b}{2})=0$. Show that$$\left|\int_{a}^{b}f(x)dx\right|\leq \frac{1}{8}(b-a)^{2}\int_{a}^{b}\left|f''(x)\right|dx.$$ I try to use Taylor's Theorem with ...
2
votes
1answer
42 views

Prove intergral inequality

If $f$ is a Riemann-integrable function on $[a,b]$ for which $\int\limits_a^b f(x) dx = 0$, and $m \leq f(x) \leq M$ for all $a \leq x \leq b$, then prove that $$\int\limits_a^b f(x)^2 dx \leq - ...
12
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1answer
303 views

Alternative proof of simple integral inequality

Problem. Let $f\in C^1(\mathbb R)$ such that $f(0) = 0$ and $0 < f'(x) \le 1$. Prove that for all $x\ge 0$ $$ \int_0^x f^3(t)\,dt \le \left(\int_0^x f(t)\,dt\right)^{\!\!2}. $$ Below is my ...
2
votes
1answer
45 views

Showing inequality in integrating polynomials

Let the polynomial $|P(x)| = a_0 + a_1x + \dots + a_nx^n$ have coefficients satisfying the relation $$ \sum_{i=0}^{n} a_i^2 = 1.$$ Prove that $$\int_{0}^{1} |P(x)| \ dx \leq \frac{\pi}{2}.$$ Show ...
0
votes
1answer
50 views

If $f \le g$ and f, g are integrable, decreasing functions, then$\int_{x}^{\infty} f \le \int_{x}^{\infty} g$?

If $f \le g$ and $f, g$ are integrable, decreasing functions, then $\int_{x}^{\infty} f \le \int_{x}^{\infty} g$? Intuitively, I suppose it holds, but I have not found any such theorem in the ...
1
vote
1answer
71 views

(Putnam) Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and

The following is a Putnam math competition problem: Let $f:[1,3] \rightarrow \mathbb{R}$ such that $-1 \leq f(x) \leq 1 $ for all x and $ \int_{1}^{3}f(x)dx = 0 $. What is the max value of ...
2
votes
1answer
60 views

$0<\int_0^\infty\frac{\sin t}{\ln(1+x+t)} dt<\frac{2}{\ln(1+x)}$

This is my first time posting so please excuse me if I don't follow the proper etiquette. This one is a rather hard problem that was assigned to me for my calculus 2 class. Thank you for your help! ...
1
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1answer
32 views

Showing f(0) is bound above by geometric mean of supremum over intervals?

So I am working on the following problem. Suppose that $f$ is entire and $n$ is a fixed positive integer. If $$I_k:=\left[\frac{2(k-1)\pi}{n},\frac{2k\pi}{n}\right],$$ for $k=1,2,\dots,n$ and ...
2
votes
1answer
41 views

Integral inequality similar to Hardy's

I am trying to solve following puzzle: We are given functions $f$, where $f(x) > 0$ and $F := \int_0^x f(t) dt$ and some real $p>1$. Does $\int_0^\infty f(x)^p e^{-x}dx < \infty$ imply ...
2
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1answer
45 views

Bound on a process satisfying certain integral and differential inequalities.

Suppose a non-negative process $x$ satisfies the following integral and differential inequalities: $$ x_t+C\int_0^tx_s^2ds\,\,<\,\,x_0+\delta+bt, $$ $$ \dot{x}_t\,\,<\,\,Kx_t^2+a, $$ where ...
1
vote
1answer
26 views

Using Gronwall's Inequality with Random Variables

Currently, I've been working with an SDE and trying to get a bound on moments. I have it down to something of the following form: $$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$ ...
2
votes
1answer
63 views

A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
0
votes
0answers
20 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
0
votes
1answer
22 views

For given mean $\mu$ of random variable X in [0,1], what is the probability distribution function $p(X)$ that makes $VAR(X)$ maximum?

Given the conditions $\int_{0}^{1} p(x)dx=1$, $\int_{0}^{1} xp(x)dx=\mu$ and $p(x)\ge0$ for $\forall x \in [0,1]$, What probability distribution function $p(x)$ makes $Var(X)$=$\int_{0}^{1} ...
0
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0answers
28 views

I attempt to combine Abel's summation with Hardy's inequality

Let $a(n)$ be a sequence of real numbers, $A(x)=\sum_{n\le x}a(n)$, with $A(x)=0$ if $x<1$ and $G(x)=\int_0^x g(t)dt$, with $g(t)\ge 0$ integrable on $[0, \infty)$, $p>1$ (is a requeriment for ...
0
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0answers
16 views

Step of proof Hardy's Inequality [duplicate]

I trying to prove the Hardy's Inequality, by Evans book. I need of a little help in the step: If $u\in L^1(B(0,r))$ satisfying $$(2-n)\int_{B(0,r)}\frac{u^2}{|x|^2}dx=2\int_{B(0,r)}u ...
11
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2answers
235 views

Easier ways to prove $\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$

Prove that $$\int_0^1 \frac{\log^2 x-2}{x^x}dx<0$$ One way to do this is use the idea in the proof of Sophomore's dream. We have $$x^{-x}=\exp(-x\log x)=\sum_{n=0}^\infty\frac{(-1)^nx^n\log^n ...
0
votes
2answers
52 views

How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt $

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
2
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2answers
59 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
6
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1answer
118 views

Seeking a More Elegant Proof to an Expectation Inequality

Let $X$ and $Y$ be i.i.d. random variables, and $\mathbb E[|X|]<\infty$, prove that $$\mathbb E[|X+Y|]\geq\mathbb E[|X-Y|].$$ This question is a re-posting of An expectation inequality. I can ...
0
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0answers
18 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
1
vote
1answer
53 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
-1
votes
1answer
20 views

I need a function for the following equality [closed]

I need an example that there exists a measurable non-negative function $f_n:X\to\mathbb{R}$ which uniform converges to $f:X\to\mathbb{R}$, and $\displaystyle\lim_{n\to\infty} \int_X f_nd\mu$ exists, ...
3
votes
1answer
122 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
0
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2answers
70 views

Can we show this inequality? (PDE question)

I am attempting to show that $$\int^t_0 \left[e^{-\lambda \kappa (t-s)} a(s)\right] ds \leq \int^t_0 \frac{\lambda}{2\kappa} a(s)^2 ds \tag{1}\label{1}$$ for any $t$, where $\lambda, \kappa >0$, ...
0
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0answers
17 views

how to separate expectation of the product of two r.v.s

Here are two non-negative random variables $X,Y$. $X$ has finite moment of order 2, and it could have infinite moments of higher order. $Y$ has finite moment of any order. They are correlated, but ...
3
votes
1answer
74 views

Estimate the integral of $(1+x^2)^{-\alpha}$, where $\alpha>1/2$

I'm reading a proof of a theorem, and there's one step I couldn't understand why. It said that for all $a>0$ and $\alpha>1/2$, $$ \int_{a}^{\infty}(1+x^2)^{-\alpha} \ \mathrm dx ...
0
votes
1answer
37 views

Sobolev/Lebesgue norm estimates in $\mathbb{R}^3$

I'm currently working on a project in which I have to establish some estimates for some global Sobolev and Lebesgue norms. We know that if we have a bounded domain $\Omega$, then for any $q \leq p^*$ ...
1
vote
1answer
34 views

An integral inequality involving increasing function

Let $0\leq a< b \leq \pi/2$ Let $f:[a,b]\to\mathbb R$ be a positive, increasing function. Prove that $\left|\int_a^b f(t)\cos(t)dt\right|\leq f(b)(b+\sin(b))-f(a)(b+\sin(a))$ I ...
2
votes
4answers
267 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
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vote
0answers
33 views

“Transference” argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
0
votes
2answers
73 views

How can I prove this inequality? $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ [closed]

Given that $a,b$ are real numbers. How can one show that $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ ?!!
30
votes
2answers
657 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 ...
1
vote
1answer
205 views

Evans PDE: Chapter 5, Problem 9 - Clarification

I've been trying to work out the solution to Question 9 in Chapter 5 of Evans, and I'm having some difficulties. I've been looking at the solution posted here: question 9 - chap 5 evans PDE And I ...
0
votes
2answers
49 views

Misunderstanding inequalities of integrals

We have to prove the following inequalities: 1) to show that $\frac{2x}{\pi }<sin\left(x\right)<x,\:and\:after\:1-e^{-\frac{\pi }{2}}\le \int _0^{\frac{\pi ...
0
votes
1answer
35 views

Obtain the triangle inequality from the Minkowski inequality

I want to prove: $$\|f-h\|_p \leq \|f-g\|_p + \|g-h\|_p$$ Minkowski inequality: $$\|f+g\|_p \leq \|f\|_p + \|g\|_p$$ Is $\|f-g\|_p \leq \|f\|_p + \|g\|_p$? It looks like we have: $$\left(\int_a^b ...
-2
votes
1answer
67 views

Why $I_n\le log\left(2\right)$? [closed]

Okay, we have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}$, and we need to prove that: 1)$I_n\le log\left(2\right)$ $,\:\:\:\:\:$ why just log(2) ? can not be 1? ...
0
votes
1answer
35 views

Showing that a function is L1

I have been struggling with this problem; it should just use some basic inequalities, but having difficulty getting them in the right order. Let $f \in L^2(\mathbb{R})$ such that it is also the case ...
1
vote
2answers
93 views

Inequality very difficult to show

1) $\int _0^1\:\frac{x^n}{x^n+1}dx\ge \int _0^1\:\frac{x^{n+1}}{x^{n+1}+1}dx$ but I dont want to use $I_{n+1}-I_n$ 2) How we can prove with direct comparison test for ( Improper ) Integrals that is ...
1
vote
1answer
46 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...
1
vote
4answers
251 views

Integral inequality 5

How can I prove that: $$8\le \int _3^4\frac{x^2}{x-2}dx\le 9$$ My teacher advised me to find the asymptotes, why? what helps me if I find the asymptotes?