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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2answers
44 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
votes
2answers
51 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
votes
1answer
109 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
16 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
50 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
19 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
100 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
votes
2answers
53 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
14 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
61 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
29 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
22 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
259 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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0answers
22 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
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2answers
70 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 $(a^2+1)(b^2+1)>a(b^2+1)+b(a^2+1)$ ?!!
27
votes
2answers
547 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
101 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
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2answers
45 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
30 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 ...
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1answer
66 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
33 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
83 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
43 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
246 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?
1
vote
1answer
49 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 ...
2
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0answers
31 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
0
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0answers
25 views

Step function integral inequality

I would like to prove the following inequality: $$\langle f,Id \rangle^2 \leq \langle f,1 \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i \mathbb{1}_{I(i)}(s)$, ...
4
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0answers
80 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 ...
2
votes
1answer
122 views

prove this intgeral inequality with $\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2$

Let $f$ be a twice continuously differentiable function from $[0,1]$ into $R$,Give that $$f(0)+2f(\frac{1}{2})+f(1)=0$$ show that $$\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2$$ ...
4
votes
1answer
88 views

If $\int_{[0,1]} x^k f(x) dx =1$ then $\int_{[0,1]} (f(x))^2 dx \ge n^2$

Let $k \in \{0,1,...,n-1 \}$ and $f:[0,1] \to \mathbb{R}$ be a continous function. If $\int_{[0,1]} x^k f(x) dx =1$ for all such $k$ then show that $\int_{[0,1]} (f(x))^2 dx \ge n^2$.
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0answers
11 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
7
votes
1answer
120 views

Prove an integral inequality $|\int\limits_0^1f(x)dx|\leq\frac{1-a+b}{4}M$

Let $f$ be a differentiable function on $[0,1]$ and $a,b\in(0,1)$ such that $a<b$, $\int\limits_0^af(x)dx=\int\limits_b^1f(x)dx=0$. Show that: $$\left|\int_0^{1} ...
3
votes
1answer
57 views

error when replacing sum by an integral

I have seen that quite often in analytic number theory, one wants to replace a sum by an integral and then estimate the error. I saw the following estimate but I can't understand how to prove it. ...
0
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0answers
13 views

Bounding oscillating integrals over short intervals

Let $f: \mathbb{R} \to \mathbb{C}$ be written as a product $$f(x)=\vert f(x) \vert \mathrm{exp}(i \,\mathrm{arg} f(x)),$$ and suppose that $\mathrm{arg} f$ is chosen so as to be continuous. Suppose ...
0
votes
1answer
12 views

Absolute value inequality for Pettis integral

Let $f:[a,b]\rightarrow E$ be absolutely continuous and Pettis integrable, i.e. there exists $I_f\in E$ such that $x^*(I_f)=\int x^*\circ f$ for $x^*\in E^*$. Because $f$ is absolutely continuous, ...
2
votes
2answers
92 views

Calculus inequality (easy)

I wanna prove that $$\forall x>1,\quad\int_{1}^{x} \frac{\sin(t)}{t} dt - x +1 < 0.$$ Is it true that I can rewrite the inequality as $$\int_{1}^{x} \left(\frac{\sin(t)}{t}-t \right)dt < ...
3
votes
1answer
90 views

Derivative with integral inequality proof

let $f(x)$ be second derivative on $[0,1]$,and $$f''(x)\ge 0,f'(x)<0,\forall x\in[0,1], f(0)=0,f(1)=-1$$ show that $$\int_{0}^{1}\sqrt{\dfrac{f'^2(x)+1}{|f(x)|}}dx\le 2\sqrt{2}$$
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2answers
78 views

How do I prove this trigonometric integral inequality?

If f is integrable and monotone on [a,b] then $\left |\int^b_a f(x)\cos x\,dx\right | \le 2(|f(a)-f(b)|+|f(b)|).$ I've tried integration by parts and using the integral inequality property but I'm ...
0
votes
1answer
27 views

A relationship between continuous function and its reciprocal

Let $f$ is a continuous, positive function in $[0,1]$ and let $M$ and $m$ are the maximum and minimum values of $f$ in $[0,1]$. It is easy to see that $$ ...
3
votes
1answer
56 views

Integral inequality of exponent

How to formally prove the following inequality - $$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$
6
votes
3answers
87 views

Bound on $f(0)^2$ by integrals of $f^2$ and $(f')^2$ on $[0,1]$.

Let $f$ be a function which is $C^1((0,1))\cap C([0,1])$. I would like to be able to show $$ \frac{1}{2}f(0)^2 \leq \int_0^1 f(x)^2dx + \int_0^1f'(x)^2dx $$ where we are assuming that $f$ is a ...
1
vote
1answer
29 views

tail inequality for expectations

I would like to upper bound the expectation $$ \mathbb{E}[X \, \textbf{1}\{X > t\}], $$ where $\textbf{I}\{p\}$ evaluates to $1$ if $p$ is true, $0$ otherwise, and $X$ is some non-negative random ...
6
votes
1answer
74 views

Inequality About $f(t)=\int_{0}^t \sqrt{\cos(x)} dx$

During my projet, I encountered the following function defined for all $\displaystyle t\in[0,\frac{\pi}{2}]$ by : $$f(t)=\int_{0}^t \sqrt{\cos(x)} dx$$ and I need to prove the inequality below : ...
0
votes
0answers
14 views

Upper-bounds to $B_z(a,b)$

Is there any standard technique to produce nice upper-bounds to the incomplete beta function $$B_z(a,b)=\int_0^z t^{a-1} (1-t)^{b-1} dt \,?$$ Disclaimer: this question is intentionally not too ...
2
votes
0answers
38 views

one inequality involving two stochastic processes

I am having trouble proving one inequality involving two stochastic processes. The problem seems simple but I just cannot handle it. Any help would be appreciated. $S_t$ and $C_t$ are two positive ...
2
votes
0answers
28 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
votes
0answers
141 views

Prove $\int_{0}^{1} |\frac {f^{''}(x)}{f(x)}| dx \geq 4$ [closed]

I find an interesting theorem,but have no idea to prove it. $f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $ Prove that if $\int_{0}^{1} \bigl|\frac ...
2
votes
0answers
60 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
3
votes
1answer
51 views

on an exponential inequality

I'm working on proving the following inequality:$$((k/2)!)^2k^{-k}\ge(2e)^{-k}$$ for any positive even integer $k$. I can use Stirling formula to prove it for large $k$'s, but I want a proof that ...
2
votes
1answer
157 views

Reference or proof for an integral inequality

The following seems believable and quasi-intuitive to me, but it also doesn't quite seem trivial, and I'm not sure whether I've seen it stated before. Let $f$ be a complex-valued integrable function ...