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

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1answer
29 views

Prove uniqueness theoremn via Gronwall inequality

A question says: Prove Theorem 1.7 (Uniqueness). Hint: suppose that $x$ and $x^*$ are distinct solutions to the same IVP (from the same initial point). Consider the function ...
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0answers
40 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
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1answer
77 views

How can show $\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2}$

I was working on a problem and reduced it to showing the following inequality: ‎‎ $$\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2};\quad ...
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0answers
18 views

Upper bound for integral on some environment of zero

I'm trying to proof an estimate that should not be too hard to proof. Let $f$ be some integrable non-negative function and $c>0$ some arbitrary constant. I claim that there exists some ...
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1answer
43 views

Where is that half coming from?

A few lessons ago, my professor proved Poincaré inequality in the following form: Let $\Omega$ be a domain contained in $\mathbb{R}^{N-1}\times(0,a)$ for some $N\in\mathbb{N},a>0$. Then for all ...
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1answer
93 views

Show $\left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx$

Given that $f: [1,e] \to \mathbb{R}$ is a continuous function, show $$ \left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx $$ My Attempt: At first it looked rather like a ...
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3answers
180 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
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1answer
23 views

Calculate limit value of sequence by inequation with integrals

I have to calculate the limit of the sequence $a_n := \sum_{k=1}^{n} \frac{1}{n+k}$ . To do so, I have to show that the following inequation is true: $\int_{n+1}^{2n+1} \frac{dx}{x} \leqslant a_n ...
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1answer
63 views

Proving an inequality involving integrals

Let $0<a<1$, $0<b<1$, $c>0$ and $d>0$, prove the following inequality: $$\frac{1}{\frac{1}{a}+\frac{1}{b}}\geq \int_{0}^\infty\frac{1}{\frac{1}{ac\exp(-cx)}+\frac{1}{bd\exp(-dx)}}$$ ...
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1answer
26 views

An integral inequality with sequence

My works lead to the true of the following inequality: For any $p>0$, there exist a constant $C_p>0$ which depends only on $p$, such that for any nonnegative sequence $(x_k)_{k\ge1}$ and for ...
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2answers
54 views

Show that $\int_{|x|<R}\frac{|f(x)|}{|x|}\mathrm dx=o(R),\quad R\to\infty.$

Suppose $f\in L^3(\mathbb R^3)$. Show that $$\int_{|x|<R}\frac{|f(x)|}{|x|}\mathrm dx=o(R),\quad R\to\infty.$$ First, I try to show that for a fixed $R_0$, ...
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1answer
22 views

Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$

$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$. Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
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1answer
31 views

How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$?

$\Omega\subset \mathbb R^n$ is bounded and open. $u,v\in H_0^1(\Omega)$. $Du$ is gradient of $u$. How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?
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1answer
41 views

Weak Law of large numbers involving a sequence and random variable

During one of our Information theory classes, the professor constructed the following set: $$T_\delta = \left\{\mathbf{y} \in \mathbb{R}^n: \frac{\sum_{i=1}^ny_i^2}{n} \leq P + ...
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1answer
49 views

Young's inequality.

I am refering to the inequality: https://en.wikipedia.org/wiki/Young%27s_inequality The standard version for increasing functions. I read the article of Young and also a generalization of this claim ...
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1answer
42 views

A function given as an integral is uniformly continuous provided the integrand is uniformly continuous

I need to show that this inequality holds: $| \int_{0}^{1} (h \nabla f(x+sh-y) -h \nabla f(x-y)) ds | \leq |h| \varepsilon(|h|)$ For a function $\varepsilon$ which verifies $\varepsilon (|h|) ...
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2answers
76 views

Prove a function is in $L^2[0,1]$

If $f\in L^2[0,1]$, and $$g(x)=\int_0^1\frac{f(t)\mathrm dt}{|x-t|^{1/2}},\quad x\in[0,1],$$ show that $\|g\|_2\le2\sqrt2\|f\|_2$. I tried Minkowski's integral inequality (with $p=1/2$, so ...
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1answer
56 views

Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Is it possible to evaluate or at least to estimate the following integrals? $$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$ and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$ I ...
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1answer
23 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = ...
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2answers
33 views

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, proving $WZ_n\rightarrow 0$

Given $Z_n\rightarrow 0$ in probability and $W$ a random variable, I need to prove $WZ_n\rightarrow 0$. I was given a hint: show that for every $\delta,\epsilon<0$ we have ...
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1answer
57 views

Prove this integral inequality

Prove this assuming $f$ is integratable: $$\int_{-\pi}^\pi\vert f(t)\vert \, dt\leq \sqrt{2\pi}\sqrt{\int_{-\pi}^\pi\vert f(t)\vert^2}\, dt =2\pi \Vert f\Vert.$$ I tried to square both sides and use ...
7
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1answer
84 views

Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove, as an auxiliary step, the following deceptively simple inequality: $$E(X^a) E(X^{a+1} \ln X) > E(X^{a+1})E(X^a \ln X) $$ where ...
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1answer
20 views

Integral Inequality calculating operator norm

I was looking at this problem: Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$ and was confused about the step with the integral inequality: $$\left|\int_{-1}^0f(t)\ ...
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0answers
23 views

Singular Gronwall type inequality

I am looking for a proof or refrence for the following Gronwall-type inequality: Let $ \varphi (t,s) $ is a continous function for $0 \leq s < t \leq T$. If the following inequality holds: $$ ...
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1answer
60 views

Please explain about Chebyshev's inequality?

I am studying Probability theory. I met Chebyshev's inequality chapter. I understood that Chebyshev's inequality is $$ P\left({|X-\mu| \ge \epsilon}\right) \le \frac{Var(X)}{\epsilon^2} $$ And I ...
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1answer
34 views

Show something converge to infinity strongly

Show that if $ Var(Y_n)=1 $ and $ \mu_n\to \infty $ quickly enough so that $ \sum_{n=1}^{\infty}\frac{1}{\mu_n^2} $ is finite, then $ p[Y_n\to \infty]=1. $ I know that I need to use Chebyshev's ...
7
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1answer
2k views

A function of two cumulative probability distributions with same first 2 moments

Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment (hence ...
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8answers
191 views

Prove that $\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$

Let $f:[a,b]\to\mathbb{R}$ be continuous and increasing, show that $$\int_a^bxf(x)dx\geq\frac{b+a}{2}\int_a^bf(x)dx$$ I am thinking of using integration by parts. First let $$F(x)=\int_a^xf(t)dt$$ ...
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21 views

Inequalities with the Integral Test

a) Use the proof of the integral test to show that $\ln(n!)\ge n\ln(n)-n+1$ for $n>1$ b) Use part (a) to show that $\ln(n!)\ge n\ln(n)$ for $n\ge 10$ I was able to solve part a) but not ...
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1answer
65 views

How to show that $S_{n+1}>S_n$?

Let $S_n = \int_0^{\pi /2}(x\sin{x})^ndx$ for all integer $n\geq1$. I want to show that $S_{n+1}>S_n$ for all integer $n\geq1$. Since there is a real $0<a<\frac{\pi}{2}$ such that ...
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0answers
13 views

An inequality between functions in Munkenhoupt class

I'm currently studying the Munkenhoupt class (also called $A_p$ class), and I'm stumbled upon the proof of the following property: Given $1 < p <\infty$, and $w\in A_p(R^n)$ (note that we have ...
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0answers
10 views

An Integral Inequality over the space of Probability Distribution with a Parameter

Suppose $P$ is the set of functions where $p\in P: R^{+2}\to R^+$ and $p(t,s)$ is differentiable in $t$. $\forall t, p(t,\cdot)$ is a probability distribution on the positive axis $s\in [0,\infty)$, ...
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1answer
71 views

Poincaré constant for a ball (circle)

I've been recently looking for a best possible Poincaré constant for a particular domains $\Omega$ (it's related to my previous question Unique weak solution to Helmholtz equation on a square) for ...
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1answer
37 views

Inequality for Sobolev fractional spaces

I recall that the Fourier transform of a function $f \in L^1 (\mathbb{R})$ is defined by $$\hat{f}(\xi) = \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}} f(x) e^{- i x \xi} \, dx.$$ We can define that ...
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1answer
59 views

Proving an Integral Inequality using the Cauchy-Schwarz inequality

Assuming Cauchy Schwarz inquality as follows... $$\left|\int_a^b{f(x)g(x)dx} \right|\le \left(\int_a^b{|f(x)|^2}dx\right)^{1/2}\left(\int_a^b{|g(x)|^2}dx\right)^{1/2} $$ Where $g(x)=0$ and ...
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1answer
16 views

Unique weak solution to Helmholtz equation on a square

I've recently started studying the modern theory of PDEs. I studied some basic properties of Sobolev space and then started with linear elliptic PDEs. I consider the following problem: For which ...
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1answer
55 views

An inequality of integrals

Let $f \in L^{2}(\mathbb{R})$ be continuously differentiable on $\mathbb{R}$. I am trying to show the following: $( \int |f|^{2} dx)^{2} \leq 4 ( \int |xf(x)|^{2} dx) ( \int |f'|^{2} dx))$. My first ...
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1answer
26 views

Integration Inequality for unbounded vs bounded function

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$ and $g:\mathbb{R}^d \rightarrow \{0,1\}$. $f$ is unbounded. Is the following true? $$\int_{x \in \mathbb{R}^d} f(x)\ \mathsf dx \geq \int_{x \in ...
2
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1answer
50 views

An inequality for $\int_{\frac{\pi}{2}}^{\pi}\frac{\sin x}{x}\ \mathrm{d}x$

Why $\dfrac{\sqrt{3}}{8}+\dfrac{1}{10}\leq\displaystyle\int_{\frac{\pi}{2}}^{\pi}\dfrac{\sin x}{x}\ \mathrm{d}x$ ?
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0answers
60 views

Integral inequality of transformed integrand with second order stochastic dominance flavor

Let $f,g : [0,1] \rightarrow [0,1]$ be two functions such that for all $x \in [0,1]$ $\int_0^x f(t) dt \geq \int_0^x g(t) dt$ and $\int_0^1 f(t) dt = \int_0^1 g(t) dt.$ Can I conclude that ...
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2answers
38 views

Prove the following integral inequality.

For $k(s)\geq0$ and $\delta\geq0$, show that the inequality $$ \delta+\delta\int_{\tau}^{t}k(s)\exp\biggl(\int_{\tau}^{t}k(r)dr\biggl)ds\leq \delta\exp\biggl(\int_{\tau}^{t}k(s)ds\biggl)$$ ...
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1answer
29 views

Kind of Cauchy-Schwarz inequality

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Define the Hilbert space $$ H(div;\Omega):=\{u\in (L^2(\Omega))^3:\nabla\cdot u\in L^2(\Omega)\} $$ equipped with the graph norm $$ ...
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1answer
89 views

Proof that $-\log \Big(\sum_y(\sum_{x}P(x)Q(y\mid x)^{\frac{1}{1+r}})^{r+1}\Big)$ is increasing

I want to prove that the following is increasing: $-\log \Big(\sum_y \sum_{x_1}P_1(x_1)Q_1(y\mid x_1)^{\frac{1}{1+r}}(\sum_{x_2}P_2(x_2)Q_2(y\mid x_2)^{\frac{1}{1+r}})^r\Big)$ Here $P_1, P_2$ are ...
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2answers
34 views

Showing Minkowski integral inequality with $p = 2$.

I have shown: $$\bigg(\int_{0}^{1}f(t)g(t)dt\bigg)^{2} \leq \int_{0}^{1}g(t)^{2}dt\int_{0}^{1}f(t)^{2}dt$$ and now I'd like to use this to show the Minkowski inequality for $p=2$, i.e. ...
0
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1answer
39 views

Integral inequality, Hölder-type, any idea?

Is it possible for a number $\alpha>1$ to have such a kind of inequality? $$\int_A f^{-\alpha} \leq C \left( \int_A f\right)^{-\alpha},$$ where the measure of $A$ is finite $A$ and $f\geq 0$ is a ...
2
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1answer
67 views

Showing an inequality with integration

I'm having trouble establishing these inequalities without using Cauchy-Bunyakovsky-Schwarz or Minkowski inequalities. Does anyone have a helpful hint? $f$ and $g$ are continuous on $[0,1]$: $$ (a) ...
2
votes
1answer
42 views

Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $

If $f$ is real-valued and continuously differentiable on $\mathbb{R}$, prove that $$ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $$ Attempt: I tried the ...
1
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0answers
15 views

Is $\int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n \geq cq^n$?

Let $f\colon (1,2)\times (1,2) \to \mathbb{R}$ be a Lebesgue measurable, bounded and non-negative function such that $$ \int_1^2 f(x,y) dy = 1, \qquad x \in (1,2). $$ Moreover, assume that for any ...
3
votes
1answer
66 views

Prove that it is impossible to bound the integral of a product by the product of integrals

How to prove that there is no $c \ge 0$ such that $$ \Big| \int_{0}^{1} f(x)g(x) dx \Big| \le c \int_{0}^{1} |f(x)| dx \cdot \int_{0}^{1} |g(x)| dx $$ where $f,g: [0,1] \to \mathbb{R}$ are ...
2
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0answers
185 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...