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

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5
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1answer
425 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
0
votes
0answers
30 views

Proving an inequality involving integrals?

I am trying to prove that $$[\sum_{i=1}^{n}(\ln t_i)^2 t_i^\alpha+A^{\prime \prime}(\alpha)][\sum_{i=1}^{n}t_i^\alpha+A(\alpha)]\ge[\sum_{i=1}^{n}(\ln t_i) t_i^\alpha+A^{\prime}(\alpha)]^2$$ where ...
0
votes
1answer
25 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
7
votes
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 ...
0
votes
1answer
47 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
1
vote
1answer
32 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 ...
1
vote
0answers
41 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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
1answer
63 views

Proving an inequality involving integrals

Let $0<a<1$, $0<b<1$, $c>0$ and $d>0$, prove the following inequality: $$\frac{1}{\frac{1}{a}+\frac{1}{b}}\geq \int_{0}^\infty\frac{1}{\frac{1}{ac\exp(-cx)}+\frac{1}{bd\exp(-dx)}}$$ ...
25
votes
1answer
476 views

Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$

Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: ...
1
vote
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 ...
2
votes
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 ...
2
votes
2answers
157 views

Given $\int_0^1 x f(x)dx=0$, show that $\int_0^1|1-f(x)|dx>1/2$

I have seen this statement before, and I would like to use it in a proof I am working on. I do not quite remember the condition on $f$--whether it is just integrable or continuous. Can someone point ...
6
votes
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 $g$ ...
0
votes
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 ...
0
votes
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 $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?
7
votes
2answers
77 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 ...
0
votes
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 ...
1
vote
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 + ...
3
votes
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 $R_0$, ...
0
votes
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 $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
1
vote
1answer
57 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 ...
1
vote
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 ...
1
vote
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|) ...
0
votes
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 ...
7
votes
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 ...
1
vote
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 ...
0
votes
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 = ...
0
votes
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 ...
0
votes
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)\ ...
3
votes
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 ...
1
vote
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: $$ ...
9
votes
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 $$F(x)=\int_a^xf(t)dt$$ ...
2
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
0
votes
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 $s\in [0,\infty)$, ...
1
vote
1answer
73 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 ...
1
vote
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 ...
2
votes
1answer
60 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 ...
2
votes
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 ...
0
votes
1answer
27 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
votes
1answer
50 views

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

Why $\dfrac{\sqrt{3}}{8}+\dfrac{1}{10}\leq\displaystyle\int_{\frac{\pi}{2}}^{\pi}\dfrac{\sin x}{x}\ \mathrm{d}x$ ?
0
votes
0answers
61 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 ...
2
votes
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 ...
1
vote
2answers
38 views

Prove the following integral inequality.

For $k(s)\geq0$ and $\delta\geq0$, show that the inequality $$ \delta+\delta\int_{\tau}^{t}k(s)\exp\biggl(\int_{\tau}^{t}k(r)dr\biggl)ds\leq \delta\exp\biggl(\int_{\tau}^{t}k(s)ds\biggl)$$ ...
0
votes
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 $$ ...
0
votes
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. ...