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

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14 views

How to verify that the following function is monotone increasing?

$\displaystyle f(x)=x\cdot\left(1-\frac{C_Bx^{B}}{\sum\limits_{k=0}^{B}C_kx^k}\right)$, where $0<x<1$, $\displaystyle C_k=\binom{n+k}{k}$, $n,B$ are integers, then, how to verify that $f(x)$ ...
0
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1answer
28 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 ...
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 $\,\, ...
1
vote
1answer
183 views

If the weighted $L^p$ norm of a measurable function is finite, is the weighted $L^p$ norm of the antiderivative also finite?

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} \,dx < \infty. $$ Define $g \colon \mathbb{R} \rightarrow ...
0
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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 ...
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0answers
34 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 ...
7
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1answer
124 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
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1answer
70 views

Inequalities help! [closed]

Let $a,b,c > 0$ and $a + b + c = 1$. Prove: $$\sqrt{\frac{ab}{c + ab}} + \sqrt{\frac{bc}{a + bc}} + \sqrt{\frac{ac}{b + ac}}\leq \frac32$$
2
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1answer
41 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
3
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4answers
184 views

Prove with integration the inequality $e(\frac{n}{e})^n < n! < n \times e(\frac{n}{e})^n$

Prove with integration the inequality, I need some advice about how to start prove it. I know that if function is Monotonically increasing function so : $$ f(1)+\int^n_1f(x)dx\leq ...
2
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1answer
41 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 - ...
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
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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 ...
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1answer
301 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 ...
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2answers
113 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
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1answer
91 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 ...
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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 $\lambda, \kappa >0$, ...
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$ ...
2
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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 ...
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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
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1answer
60 views

How to prove the inequality in Burgers equation and what is the relationship to energy dissipation

Consider Burgers equation: \begin{equation} u_t+uu_x-\epsilon u_{xx}=0 \quad , \quad u(x,0)=g(x) \quad , \quad u = 0 \mbox{ when } |x| \mbox{ large}, \end{equation} where ...
1
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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
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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! ...
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 ...
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2answers
62 views

Inequalities for combinations of $\int f $ and $\int (1/f)$ where $m\le f\le M$ on an interval

Let $f\in C[a,b]$. Assume that $\min_{[a,b]}f=m>0$ and $M=\max_{[a,b]}f$. Which one is true? a. $$\frac{1}{M}\int_a^bf(x)dx+m\int_a^b\frac{1}{f(x)}dx\geq 2\sqrt{\frac{m}{M}}(b-a)$$ b. ...
6
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3answers
295 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 ...
2
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1answer
67 views

Cauchy Schwarz with integrals of integrable functions

I was reading and doing problems from Spivak's Calculus on Manifolds. Q1-6 (a) stumped me a little. Let $f$, $g$ be integrable on $[a,b]$. Prove that $$\left| \int_a^b f\cdot g \; \right | \leq ...
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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
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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
97 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 ...
2
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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 ...
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
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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: $$X(t)^p \leq a(t) + \int_0^t X(s)^pY(s) ds + \int_0^t X(s)^p dW_s$$ ...
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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
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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} ...
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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 ...
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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 ...
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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 ...
0
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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 ...
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 ...
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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
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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, ...
4
votes
0answers
94 views

Inequality involving double integral

There's a function $g(x,y):\mathbb{R^+\times \mathbb{R}^+\rightarrow \mathbb{R}^+}$ with $g_1(x,y)>0$, $g_2(x,y)>0$, and $g_{12}(x,y)>0$. I conjecture that $$\int ...
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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0answers
16 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
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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 ...