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

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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 ...
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2answers
29 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 ...
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1answer
24 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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48 views

Nobody can help me ? I can't believe that…

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? ...
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$E[R^2]$ by saprcings

If you define R as the range: R=X_(n)-X_(1) and let be the sparcings S_r=X_(r+1)-X_(r), I found in this article http://www.ime.unicamp.br/sinape/sites/default/files/article.pdf page 7, than ...
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1answer
29 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 ...
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81 views

Impossible to prove that is unbounded!

How we can demonstrate that $$\int _1^e\:\left(1+\log\left(x\right)\right)^ndx$$ is unbounded as $n\to \infty$, without using Bernoulli's Inequality?
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75 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 ...
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1answer
35 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 ...
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232 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?
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45 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 ...
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29 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 $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
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24 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 $f(s)=\sum\limits_{i=1}^N f_i \mathbb{1}_{I(i)}(s)$, ...
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1answer
113 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 $$\int_{0}^{1}(f''(x))^2dx\ge 1920\left(\int_{0}^{1}f(x)dx\right)^2$$ ...
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85 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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9 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 ...
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1answer
111 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} ...
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1answer
56 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. ...
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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 ...
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1answer
11 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, ...
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89 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 < ...
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1answer
88 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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70 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 ...
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1answer
25 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 $$ ...
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1answer
54 views

Integral inequality of exponent

How to formally prove the following inequality - $$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$
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85 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 ...
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1answer
28 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 ...
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1answer
73 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 : ...
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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 ...
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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 ...
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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 ...
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139 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 ...
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56 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 ...
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1answer
49 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 ...
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1answer
156 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 ...
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1answer
28 views

how to prove this equality $||f||_{L^p}^{p}=p\int_0^{+\infty} \lambda^{p -1}\mu(E^f_\lambda) d\lambda$

Let $(X,B(X),\mu)$ be a measure space, suppose there is a function f that is measurable Define the distribution function ${\mu(E_\lambda^f): {\mathbb R}^+ \rightarrow [0,+\infty]}$ How to prove ...
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82 views

Minkowski inequality of infinite sum

For $1\leq p <\infty,$ Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$ Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty ...
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1answer
47 views

Cauchy Schwartz 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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Detail in a proof about energy minimizing harmonic maps

Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional ...
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1answer
49 views

What inequality was applied?

I'm reading some probabilistic paper and have a trouble with understanding some part. Here is this part: mu is a Lipschitz function and M is the Lipschitz constant, and: What inequality was ...
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46 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. ...
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3answers
85 views

How to prove Cauchy-Schwarz integral inequality?

The Cauchy-Schwarz integral inequality is as follows: $$ \displaystyle \left({\int_a^b f \left({t}\right) g \left({t}\right) \ \mathrm d t}\right)^2 \le \int_a^b \left({f \left({t}\right)}\right)^2 ...
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An exponential integral inequality [duplicate]

Let $f:[0,1]\to \mathbb{R}^+$ be a continuous function. Prove that $$ \int_{0}^{1}f(x)\,\mathrm{d}x-\exp\left(\int_{0}^{1}\log f(x)\,\mathrm{d}x\right)\leq \max_{0\le x,y\le ...
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95 views

How to show $\int_0^1 f^3 dx < ( \int_0^1 f dx )^2 $ [duplicate]

Assume $f$ is $C^1([0,1])$ and $f(0)=0$ and $ 0 < f' \le 1 $ then I want to show that $$\int_{0}^{1} f^3 dx < \left( \int_{0}^{1} f dx \right)^2 $$ my tries : I want to use a similar way like ...
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3answers
266 views

How prove this integral inequality $4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$

Question: let $$f(0)=0,f(1)=1, f''(x)>0,x\in (0,1)$$ let $k>2$ are real numbers,show that $$4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$$ This problem is from china ...
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1answer
43 views

Does this variation of Jensen's inequality hold?

The original Jensen's inequality in probability theory is generally stated in the following form: if $X$ is a random variable and $f$ is a convex function, then $f \left(\mathbb{E}[X]\right) \leq ...
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0answers
39 views

Prove that $\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}$ [duplicate]

Let $(X,\mathcal{A}, \mu)$ be any measure space and let $u \in \bigcap_{p\in [1,\infty]} \mathcal{L}^p(\mu)$. Then $$\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}.$$ I have already proved the ...
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1answer
48 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 ...
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1answer
61 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 ...
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2answers
71 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 ...