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Questions tagged [integral-inequality]

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

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35 views

Integral inequality with some parameters

I'm studying a paper and I'm not able to prove the following inequality: $$\int_{U(r)} \frac{1}{(2\pi)^{k+1}}\prod_{j=1}^{k+1}\bigg(2\frac{\sin(b(x_j-y_j))}{x_j-y_j}\bigg)^2\,d\sigma(y) \leq c(k)b^{2(...
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1answer
44 views

under which situation inequality between two integral yeilds pointwise inequality?

suppose I have $$\int{f(x)} < \int{g(x)}$$ when I can conclude this : $f(x_{0})< g(x_{0})$ for some $x_{0}$ or is there such a $x_{0}$? how I can find it? if there is not answer, how ...
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32 views

About an integral inquality

Set $\phi (x) = u(x)+iv(x)$, $x=(x_1,...,x_N)$, a $T$-periodic function in $H^1_{loc}(\mathbb{R}^N)$, that is $\phi (x) = \phi (x_1 + T ,..., X_N + T)$ for all $x$ and where $u = Re\ \phi$ and $v = ...
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1answer
34 views

Proving that $\int_0^1 \left(\frac{\partial T}{\partial z}(t,z)\right)^2\mathrm{d}z \geq 2 \int_0^1 T^2(t,z)\mathrm{d}z$

Exercise : Assume that $T$ satisfies the equation $T_t(t,z) = aT_{zz}(t,z)$ for $t>0, z \in (0,1)$ and $a > 0$ a constant. Moreover, suppose that $T(0,z) = T_0(z)$ for $z \in [0,1]$, where $...
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2answers
77 views

Let $f:[0,1]\to[1,3]$ be continuous. Prove $1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}$

Let $f:[0,1]\to[1,3]$ be continuous. Prove $$1 \leq \int_0^1 f(x)\,\mathrm dx \int_0^1 \frac{1}{f(x)}\, \mathrm dx\leq \frac{4}{3}.$$ The left is just Cauchy's inequality with integral form, but ...
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41 views

differential inequality involving p.s.d matrix

Problem description Let $h(t) > 0$ be a continuous real function and $x(t) \in \mathbb{R}^{3}$ be also a continuous function. Let $$T(t) = h + \dot{x}^T\dot{x}$$ It is known that $$ \dot{T} = -\...
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0answers
30 views

Conditions to ensure an integral to be positive

Suppose that $g:[0,1]\to\mathbb{R}$ is a differentiable function with $g(0)=0$. I want to find conditions on $g$ such that $$ \int_0^1 f(x)g(x)\int_x^1f(t)\,dt\,dx=\int_0^1f(t)\int_0^xf(x)g(x)\,dx\,...
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27 views

Minimizing the value of integral under certain conditions

I have been working on the following minimization problem: Minimize $I(f) := \int_{-\infty}^{\infty}{|f(x)|^2+|f'(x)|^2dx},$ where $f\in\mathcal{A} = \{f:\mathbb{R}\to\mathbb{R}$ is differentiable ...
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1answer
39 views

The relation between the square of the integral and the integral of the square of the integrand

Let $Y$ be a $\sigma$-finite measure space and $f$ be a $L^2$ function on $Y$. Then does the following formula always hold? $$\left| \int_{Y} f\right|^2 \leq \int_{Y} |f|^2$$ I know that if $Y$ is a ...
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1answer
35 views

Prove or disprove following inequality of integrals [closed]

If $p$, $q$ are positive numbers such that $p<q$, prove that $$\int_{0}^{\infty}\frac{dx}{1+x^{p}}> \int_{0}^{\infty}\frac{dx}{1+x^{q}}.$$
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Prove $(E|Z^r|)^{1/r} \geq (E|Z^s|)^{1/s}$ via log-convexity

Please help with the following: Using the fact that $p\mapsto \log E|Z^p|$ is a convex function of $p$, show that for any $r\geq s >0$ that $$ (E|Z^r|)^{1/r} \geq (E|Z^s|)^{1/s}.$$ I don't know ...
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17 views

An integral inequality on a disc about twice derivatives.

Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\} $, where $\partial D=\{(x,y):x^2+y^2=1\}$, ...
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14 views

About the sufficient conditions for an upperbound for an integral ratio.

Let f and g be functions, I am interested in finding (if there is any) the sufficient conditions on f and g such that we can satisfy an inequality of this type: $$\frac{\int_{0}^{\infty}f(x)dx}{\int_{...
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1answer
40 views

Maximum of $\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$

Let $p>q>0$ and $C=\{f:[0,1] \to \mathbb{R} \mid f \text{ is continuous} \}$. Determine $$\max_{f \in C}\int_0^1(x^p|f(x)|^q-x^q|f(x)|^p)dx$$ and the functions for which this maximum occurs. ...
2
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1answer
88 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
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1answer
54 views

$f,g \in [0,1] \times [0,1]$, $\int f - g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

$f,g$ are monotonically increasing in $[0,1]$ and $0\le f , g \le 1$. $\int_0^1 f - g \mathrm{d}x = 0$. Prove that $$\int_0^1 |f - g|\mathrm{d}x \le \frac{1}{2}$$ In my previous question, $g(x) = x$....
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82 views

Proof-Verification: $ \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x)\,{\rm d}x \leq \frac{1}{2019}\int_a^b f(x)\,{\rm d}x$

Problem Let $f(x)$ be continuous and increasing over $[a,b]$. Prove $$\displaystyle \int_a^b \left(\frac{b-x}{b-a}\right)^{2018}f(x){\rm d}x \leq \frac{1}{2019}\int_a^b f(x){\rm d}x.$$ Proof By ...
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1answer
59 views

Is it always true that integral of nonnegative function is non negative [closed]

If $f(x)\geq 0$, is it true that $\int f(x)dx \geq 0$?
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1answer
56 views

Integral inequality $\int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(0)=0$. $$\text{Prove that} \qquad \int_{0}^{1}\left ( \frac{f(x)}{x} \right )^2dx \leq 4\int_{0}^{1}(f'(x))^2dx$$ My attempt: Let $$g(x)=\begin{cases} ...
1
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1answer
44 views

Integral inequality issue

Given that the bilinear form on the set $$V:= \{v\in C^2[0,1] v(0)=0=v(1)\}$$ is defined as $$[u,v]=\int_0^1 [pu'v'+quv]dx,$$ where $p\in C^1[0,1], p(x)\ge p_0>0, q(x)\ge 0, q\in C[0,1]$, I want ...
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2answers
74 views

Integral inequality $\int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$

Let $f \in C^1([0;1],\mathbb{R})$ such that $f(1)=0$. $$\text{Prove that} \qquad \int_{0}^{1}|f(x)|dx \leq \int_{0}^{1}x|f'(x)|dx$$ My attempt: \begin{align} \int_{0}^{1}x|f'(x)|dx = \int_{0}^{1}...
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0answers
50 views

Gagliardo-Nirenberg inequality for fractional Sobolev spaces

Wikipedia states two versions of the Gagliardo-Nirenberg inequality for nonfractional Sobolev spaces. I'm interested in generalizations to fractional (Slobodeckij) Sobolev spaces. Such a ...
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1answer
48 views

Bounding integral of square root by square root of integral

Let $f(x)\geq 0$ be a function over $[0,\infty)$. How can I lower bound $\int_{x=0}^{u}\sqrt{f(x)}dx$ by $c \sqrt{\int_{x=0}^{u}f(x)dx}$ where $\sqrt{\int_{x=0}^{u}f(x)dx}<\infty$ and $c>0$ is ...
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2answers
61 views

Inequality for bounded locally integrable functions

$\textbf{The Problem:}$ Let $f\geq 0$ be a bounded function and $E\subset\mathbb R^d$ have finite measure. Prove that there exists $R>0$ such that for all $0<r<R$ we have $$\int_{E}f(x)dx\...
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2answers
191 views

$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$ for a convex differentiable function

If $f:[a,b] \to \mathbb{R}, f(a)=0,f(b)=1$ is a convex increasing differentiable function on the interval $[a,b]$ . Prove that $$\int_a^bf^2(x)\,dx\le \frac{2}{3}\int_a^bf(x)\,dx$$ Since f is ...
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2answers
89 views

Proof of maximal principle on Laplace Equation involving Poisson integral formula

This question appeared on a past PDE exam I found while studying for my finals: Let $u(r,\theta)$ be solution to the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}...
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1answer
38 views

Littlewood's inequality for $L^p$ spaces

I have tried to prove the following inequality, but I couldn't do yet. Prove the following interpolation estimate: $$\| u\|_q \leq \| u\|_p^{\theta} \| u\|_r^{1- \theta}$$ where $p≤q≤r$, $θ∈[0,1]$ ...
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1answer
69 views

Heisenberg’s inequality

I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet. Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a \in \mathbb R$ so that $f′$ ...
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1answer
113 views

Proving $\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6$

I have a question where I have to show $$\frac12 < \int_0^1 \frac1{\sqrt{4-x+x^3}} dx < \frac{ \pi }6 \approx 0.52359$$ using the result $$\frac12 < \int_0^{1/2} \frac{1}{\sqrt{1-x^{2n}}} ...
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0answers
25 views

Exponential decay and integration

I am confronted with the following problem: Let $\mu$ be a probability measure on $\mathbb{R}$. We wish to show that for any $p \in \mathbb{N}$ and $r \in \mathbb{R}$, the integral $$ F(r):= \...
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0answers
15 views

If a rotational family is close to a single rotation must its derivative be small?

Let $\mathbb D^n$ be the closed $n$-dimensional unit disk. Let $f:\mathbb D^n \to \text{SO}(n)$ be a smooth map. Let $R \in \text{SO}(n)$ be a fixed rotation. I am trying to prove a quantitative ...
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2answers
54 views

If $f(a)=f(b)=0$ and $|f''(x)|\le M$ prove $|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$

If $f(a)=f(b)=0$ and $|f''(x)|\le M$. Prove $$|\int_a^bf(x)\mathrm{d}x| \le \frac{M}{12}(b-a)^3$$ I have thought about that since $f(a) = f(b) = 0 $ there is $\xi$ such that $f'(\xi) = 0$. Then when ...
4
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2answers
54 views

If $f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s$ prove $f(t)\le 1+t$

If $f(x)$ is positive and continuous on $[0,1]$ and $f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s$, prove that $f(t)\le 1+t$. Here's my thinking. $$f^2(t) \le 1+2\int_0^tf(s)\mathrm{d}s \Rightarrow f(t)\le ...
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0answers
15 views

Can we include inequalities while solving under determined simultaneous linear equation in reduced Echelon form?

Is there a way to include the inequalities of variables in calculating family of equations while solving under determined simultaneous linear equations? For Eg: Lets say x + y = 4, y + z = 4. But ...
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1answer
34 views

Given a sequence of Lp functions, does the integral commute with the lp norm?

I have been struggling to prove the following: Let $ \{ f_n \}$ be a sequence in $ L^p(E) $ for some $ p \geq 1 $. Then, $$ \left( \sum_{n=1}^\infty | \int_E f_n \mathrm{d}\mu |^p \right)^{ \frac{1}...
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1answer
55 views

Prove that:$\int_{a}^{b}(2x^{3}-3(a+b)x^{2}+6abx)f'(x)dx\geq (a-b)^{3}f(a)$.

Let $f:[a,b]\rightarrow[0,\infty)$ a differentiable function with its derivative continuous and $f(a)=f(b)$. Prove that:$\int_{a}^{b}(2x^{3}-3(a+b)x^{2}+6abx)f'(x)dx\geq (a-b)^{3}f(a)$. I tried to ...
1
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1answer
48 views

$\int_0^1f(x)\cdot x^{n+1}\text{d}x > \int_0^1f(x)\cdot x^n\text{d}x \cdot \int_0^1f(x)\cdot x\text{d}x$

I have convinced myself that $$\int_0^1f(x)\cdot x^{n+1}\text{d}x > \int_0^1f(x)\cdot x^n\text{d}x \cdot \int_0^1f(x)\cdot x\text{d}x$$ is true whenever $f$ is non-negative, $\int_0^1f(x)\text{d}x=...
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2answers
23 views

prove inequality with an integral over region with unit length

I am trying to show that $\log{m}\le{\int_{m}^{m+1}{\log{t}}}dt$, with $m\ge{1}$. I tried simplifying the problem to $0\le{\int_{m}^{m+1}{\log{\big(\frac{t}{m}\big)}}}dt$, but can't seem to get any ...
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votes
0answers
70 views

How is the inequality of these integrals true?

I'm doing an exercise from baby Rudin (chapter 8 exercise 11) and found a suggestion that it might use. $$\left|\int_0^\infty e^{-x}f\left(\frac{x}{t}\right)dx\ -1\right| \leq \int_0^\infty e^{-x}\...
3
votes
0answers
85 views

Is it a well-known inequality?

Let $f\in \mathcal{C}^2(\mathbb{R},\mathbb{R})$ and suppose that $\int_\mathbb{R}f^2<+\infty$ and $\int_\mathbb{R}f''^2<+\infty$. Then we can deduce that : $\left(\int_{\mathbb{R}}f'^2\right)...
1
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0answers
18 views

Linear version of Gronwall's inequality, proof

I am reading a proof of the following theorem: Assume $\phi$ is a continuous function in $[0,T]$ that satisfies $$\phi(t) = \alpha + \int_0^t (\beta \phi(s) + \gamma)ds, \hspace{0.5mm} t\in [0,T], $$ ...
3
votes
1answer
30 views

Help bounding a “norm”

In Weak Convergence and Stochastic Processes, the authors introduce the following notation: $$\|\xi\|_{2,1} = \int_0^\infty \sqrt{P(\xi > x)}\,\mathrm dx$$ They then admit that this is technically ...
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1answer
137 views

Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.

Verify that $$\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$ and $r > 0$. I'm stuck. here is my attempt: $|...
1
vote
0answers
17 views

Finding discrete solutions to inequality involving Exponential Integral

I want to identify the least natural number $n$ (of course, it suffices to solve this problem for the reals, and then take the floor) such that $$-c \text{Ei}\left(-e^{\frac{a-d}{c}} (n+1)\right)+a-...
0
votes
0answers
28 views

Integral inequality with partial derivative

Fix $\eta \in \Bbb R^n$, then for any $\phi \in C_c^\infty(\Bbb R^n)$, i.e. compactly supported smooth function, $$ \int_{\Bbb R^n} \frac{\partial}{\partial x_j} (\phi(x) (x \eta)) \, dx \le \int_{\...
0
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0answers
35 views

How to find the sharp constant between norms?

How to prove that $\|u\|_\infty\leq C_p\|u'\|_{L^p},\ \forall\ u\in W^{1,p},\ u(0)=u(T)$ with $\int_0^T u=0$ and $C_p=\frac{1}{2}\left[\frac{T(p-1)}{2p-1}\right]^{\frac{p-1}{p}}$? It is easy to show ...
1
vote
1answer
183 views

Application of Gronwall Inequality to existence of solutions

Consider the $N$-dimensional autonomous system of ODEs $$\dot{x}= f(x),$$ where $f(x)$ is defined for any $x \in \mathbb{R}^N$, and satisfies $||f(x)|| \leq \alpha||x||$, where $\alpha$ is a ...
1
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1answer
35 views

Integration inequality, why can we pull e^t out of the integral and leave its e?

In the following, at line 3 $e^t \sin(t)$ is pulled out of the $|\cdot|$ and left as a constant. How does one justify this step?
1
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1answer
25 views

Inequality used in proof of existence of SDE solutions?

In the proof of existence/uniqueness of SDE the following inequality is used: $$E\left[ \left( \int_0^t a(s,\omega) ds \right)^2 \right] \leq t E\left[ \int_0^t a(s,\omega)^2 ds \right]$$ and I ...
2
votes
3answers
87 views

Compute the limit $\lim_{n\to\infty} I_n(a)$ where $ I_n(a) :=\int_0^a \frac{x^n}{x^n+1}\,\mathrm{d}x, n\in N$.

For $a>0$ we define $$\space I_n(a)=\int_0^a\frac{x^n}{x^n+1}\,\mathrm{d}x , n\in N.$$ Prove that $0\le I_n(1) \le \frac{1}{n+1}$ Compute $\lim_{n\to\infty} I_n(a)$ My attempt: I regard $I_n(1)=...