Questions tagged [integral-inequality]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
41 views

Does the variance of these random variables tend to zero?

Let $X$ be a probability space, and let $F:[0,\infty) \to [0,\infty)$ be a $C^2$ strictly convex function. Suppose that $F''$ is an everywhere positive strictly decreasing function, and that $\lim_{x ...
0
votes
1answer
29 views

Comparing an improper integral with the one for the same integrand multiplied by the integration variable

Consider the following improper integral, whereby $a \neq 0 $ is real and the $C^\infty$ real function $f(x)$ is further such that the following integral vanishes: $$ \int_{-\infty} ^{\infty} \frac{f(...
0
votes
1answer
46 views

Why is this inequality true?

In a paper I'm reading, they say Let $\delta$, $C_0$, and $n$ be positive constants. Then $$\frac{\int_{\delta}^{\infty} \sqrt{n u^{\alpha}} e^{-nu^2 / (2C_0)}\,du}{\lambda_0^n} < \infty$$ for ...
1
vote
0answers
10 views

Gronwall's inequality generalisation

I'm stuck trying to prove some generalisations to the Gronwall's inequality. Firstly, I saw that you can prove it in the case tha $M \in \mathbb{R} $, $u \in C^0[t_0,t_1]$ and $a \geq 0$ and Riemann ...
1
vote
0answers
102 views
+200

Functional inequality proof

Let $u = u(t,x)$ and $$ \dot{W}(t) \leq -\left(\frac{\sigma \omega \pi^2}{4L^2} + g\right)\int_0^L u^2 dx - \sigma(1-\omega) \int_0^L u_x^2 dx - aμ \int_0^L u^2_t dx $$ $$ -σ^{-1}c^2Q\int_0^L (u_t - ...
2
votes
0answers
28 views

Counterexample for Korn inequality when $p=\infty$

I am currently studying the proof of Korn inequality for $1 < p < \infty$ for an exam, and a counterexample for $p=1$. I've read in the article that my professor gave me that this result doesn't ...
0
votes
2answers
33 views

Prove that Integration of sqrt sinx from 0 to π/2 is greater than π/3 [closed]

Prove that $\int_0^{ \pi/2}\sqrt{\sin x}dx>\pi/3$ I can prove it greater than $1$ but this I don't know how to solve
1
vote
3answers
68 views

Find the maximum value of $\int_0^1 x^2f(x) - xf^2(x) dx$

given $f:[0,1] \to \Bbb{R}$, find the maximum value of $$\int_0^1 x^2f(x) - xf^2(x) dx$$ I tried to factorize it as such: $$I = \int_0^1 xf(x) ( x - f(x) )dx$$ Then tried Cauchy-Schwarz $$\int_0^1 ...
0
votes
0answers
22 views

How to derive this inequality for the integral of $exp(x)^2$?

Im looking at the solution of some exercise problem. It uses the following inequality $$\frac{1}{\sigma} \int_0^{\theta-\theta_0} \left(\exp(x/\sigma) \cdot A - \frac{1}{2\sigma}\right)^2 \,dx \leq \...
9
votes
2answers
155 views

Show $\int_0^t (t-x)P_n(x)\,dx\leq \frac{t^2}{2}\int_0^1 P_n(x)\,\mathrm dx $ where $P_n(x)=(x(1-x))^{n}$

Show that for all $t\in [0,1]$, and for any $n\in\mathbb{N}$, $$\int_0^t (t-x)P_n(x)\,dx\leq \frac{t^2}{2}\int_0^1 P_n(x)\,dx\tag{*}$$ where $P_n(x)=(x(1-x))^{n}$. Since $P_n\geq 0$ over $[0,1]$ ...
0
votes
0answers
45 views

Reduction Formulae of $I_n=\int_0^\frac{\pi}{4}\tan^nx\, dx$ and $J_n=(-1)^nI_{2n}$

Let $I_n=\int_0^\frac{\pi}{4}\tan^nx\,dx$ and let $J_n=(-1)^nI_{2n}$ for $n=0,1,2$ Show that $I_n+I_{n+2}=\frac{1}{n+1}$. Deduce that $J_n-J_{n-1}=\frac{(-1)^n}{2n-1}$ for $n\ge1$ Show that $J_m=\...
0
votes
0answers
25 views

Inequality for joint probabilities of dependent random vectors

Let $X$ and $Y$ be two dependent random vectors in $\mathbb{R}^d$, such that $X\neq Y$ with probability 1, whose joint probability measure has density $\mu(x,y)$ with respect to the Lebesgue measure. ...
1
vote
0answers
25 views

References for variants of Jensen inequality

I am looking for a reference for the following claim: Let $X$ be a probability space, and let $g:X \to \mathbb [0,\infty) $ be in $L^1(X)$. Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be convex and ...
2
votes
2answers
148 views

Wirtinger's inequality variation

If $f \in C^1[0,1]$ with $f'(0) = f(1) = 0$, then$$\|f\|_2\leq\frac2\pi\|f'\|_2.$$ Elaboration: Assume the Sturm-Liouville operator $A: D \longrightarrow L^2(0,1)$ where the domain is $$ D = \{f \in ...
0
votes
1answer
38 views

An inequality of Alternating Series

We have, $\sum_{n=1}^{\infty}\frac{ (-1)^n \sin (t \log n)}{n^\sigma}$ =0 $\sum_{n=1}^{\infty}\frac{ (-1)^n \sin (t \log n)}{n^\sigma}\leq\sum_{n=1}^{\infty}\frac{ (-1)^n}{n^\sigma}$. Since \sin(t \...
1
vote
1answer
94 views

Let $f: \mathbb R \rightarrow [0,\infty)$ be countinuously differentiable. Show that:

$$\left|\int_0^1f^3(x)dx - f^2(0)\int_0^1f(x)dx\right| \leq \max_{0 \leq x \leq1} |f'(x)| \left(\int_0^1f(x)dx\right)^2$$ here the exponents means the exponential and not the composition. My attempt ...
1
vote
1answer
26 views

Inequality Integral who involves logarithms and exponentials

Show that: $$\int_1^e (x^2+1)\log^n(x) dx \leq \frac{2}{e^{n+1}} \int_1^e e^x\cdot x^n dx$$ My approach: I tried using Cauchy-Swartz but not work. I tried $\log(x) \leq x, \forall x \geq 1$ but still ...
1
vote
0answers
25 views

A question from Stein's book, Harmonic Analysis, oscillatory integral of the second kind.

My question is about a claim in Stein's book"Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals", page 379. It says that $$|K_{\lambda}(\xi,\eta)|\leq A_N(1+\lambda|\xi-...
1
vote
2answers
37 views

Need help in proving a version of Stirling Formula

I am trying to prove this inequality but unable to do so -> $log(n!) \leq \int_2^{n+1} log(x) dx $ . My attempt -> using Stirling Formula LHS = n log(n) - n + O ( log(n) ) and Integral is n( log(n+...
-1
votes
0answers
33 views

Asking if an argument in an inequality is valid

I am asking if an argument which I am trying to use make sense or not Assume $I(z) = \int_0^1 f(x)^n 1/x^2 \, dx$ then I wrote $\lim_{n \to \infty} I(z)^{1/n} \leq \int_0^1 f(x) \, dx$ , assuming ...
2
votes
0answers
44 views

How to prove the following integral inequality?

Let $f:[0,1]\to \Bbb R$ be differentiable function with $f(0)=0$ and $0\leqslant f(x)'\leqslant 1$. Then $$3\left(\int_0^1f(x)^2dx\right)^3\leqslant \int_0^1f(x)^8dx.$$ My attempt: $0\leqslant f(...
0
votes
1answer
52 views

Equality situation in Jensen's inequality

Let $(X,\Sigma,\mu)$ be a probability space, and let $g:X \to \mathbb [0,\infty)$ be measurable. Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be a convex $C^1$ function. Furthermore, assume that ...
1
vote
2answers
52 views

Prove or disprove that $\int_a^bf(x)g(x)dx\geqslant g(b)\frac{\int_a^bf(x)dx}{b-a}$.

Let $f:[a,b]\to\mathbb{R}$ and $g:[a,b]\to\mathbb{R}$ be continuous, positive-valued functions on interval $[a,b]$ and let $g$ be decreasing. Then prove or disprove the following inequality $$\int_a^...
6
votes
2answers
127 views

Given $\int_{\frac13}^{\frac23}f(x)dx=0$, how to prove $4860(\int_0^1f(x)dx)^2\le 11\int_0^1|f''(x)|^2dx$?

Suppose $f\in C^2[0,1]$, and $\int_{\frac13}^{\frac23}f(x)dx=0$. Prove that $$\left(\int_0^1f(x)dx\right)^2\le \frac{11}{4860}\int_0^1|f''(x)|^2dx.$$ This problem is quite similar to Prove ...
4
votes
1answer
180 views

Proving that the solution to $f^{\prime}(x) = \frac{1}{x^{2} + (f(x))^{2}}$ is bounded above.

I am given that $f:[0,\infty)\to \mathbb{R}$ is the unique solution to the ODE: $$f^{\prime}(x) = \frac{1}{x^{2} + (f(x))^{2}}$$ with $f(0)=1$ and I must prove that it is bounded. I have already ...
7
votes
3answers
217 views

Finding the maximum value of $\int_0^1 f^3(x)dx$

Find the maximum value of $\int_0^1 f^3(x)dx$ given that $-1 \le f(x) \le 1$ and $\int_0^1 f(x)dx = 0$ I could not find a way to solve this problem. I tried to use the cauchy-schwarz inequality but ...
2
votes
1answer
49 views

Prove $\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4$ when $f(0)=f(1)=0$. [duplicate]

Define $f$ has second continuous derivative over $[0,1]$. And $f(0)=f(1)=0$, And $f(x)\neq0 \,when \ x\in(0,1)$, prove $$\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4,$$ I can't use $f(0)=0 \ and\...
1
vote
1answer
61 views

Prove $\int_{0}^{1}|f(x)|^2\,dx\le\int_{0}^{1}|f{'}(x)|^2\,dx$ with $f(0)=0$ [duplicate]

Let $f$ has following properties: 1)f has continuous derivative in$[0,1]$; 2)$f(0)=0$; then proof $$\int_{0}^{1}|f(x)|^2\,dx\le\int_{0}^{1}|f{'}(x)|^2\,dx$$ I think the condition "$f(0)=0$"may be ...
1
vote
0answers
25 views

A singular integral

So, I was doing some PDE related computations and I obtained the following integral $$ \iint \frac{(y-y')\,f(x',y')}{(|x-x'|^2+|y-y'|^2)^\frac{3}{2}}\, \left|\frac{x'}{x}\right|^2 \,\mathrm{d} x'\,\...
1
vote
1answer
53 views

Prove that $\int_{-a}^{b} x^2f(x)dx\leq ab\int_{-a}^{b}f(x)dx$.

Let $a,b>0$,$f\in C[-a,b]$,$f>0$,$$\int_{-a}^{b} xf(x)dx=0,$$then prove that $$\int_{-a}^{b} x^2f(x)dx\leq ab\int_{-a}^{b}f(x)dx,$$ I have tried with the $$\int_{-a}^{b} x^2f(x)dx=\varepsilon^2\...
2
votes
1answer
85 views

A math inequality cannot prove

I try to solve a inequality, I use R to run simulation with different $\alpha>0$ and a sequence of $n$, the simulation all show the LHS is smaller than RHS, but I cannot prove it analytically, can ...
1
vote
2answers
62 views

Show that $\int_0^1[1+f(x)]dx\int_0^1\frac{1}{1+f(x)}dx\le1.125$

Show that $$\int_0^1[1+f(x)]dx\int_0^1\frac{1}{1+f(x)}dx\le1.125$$ where $f:[0,1]\rightarrow [0,1]$. In the beginning, I was going to use Chebyshev's inequality but then noticed that we don't have ...
0
votes
0answers
38 views

Integral inequality proof.

How to prove this inequality ? $\displaystyle \frac{2}{\pi^2}<\bigg|\int\limits_{\pi}^{2\pi}\frac{\cos(x)-1}{x^2}\bigg|\leq \frac{2}{\pi} .$
2
votes
3answers
85 views

Show that $\int_0^1|x-\mu|f(x)dx\le \frac{1}{2}, \text { where } \mu=\int_0^1xf(x)dx.$

Question: Let $f:[0,1]\to(0,\infty)$ be a function satisfying $$\int_0^1f(x)dx=1.$$ Show that the integral $$\int_0^1(x-a)^2f(x)dx\text{ is minimized when } a=\int_0^1xf(x)dx.$$ Hence or otherwise ...
0
votes
1answer
31 views

Why does this integral inequality holds?

I am reading a paper, and the authors write the following inequality for two functions $v$, $z$, both $\mathbb{R}\to\mathbb{R}$: $$\int_{0}^{t} \left(|v(u)|^2\left(\int_0^u|v(r)|^2dr\right)\left(\...
0
votes
0answers
17 views

Question regarding the existence of a second moment (related to renewal theory)

Let $S$ denote a random walk with i.i.d. increments $X_1, X_2, \dots$ Let $U$ denote be the function $$ U(x) = \sum_{i=1}^\infty P(S_{\sigma_i} \leq x), $$ where $\sigma_i$ are the strictly ascending ...
0
votes
1answer
90 views

An integral inequality $f(t)+\int^{\infty}_tg(s)ds\leq Ce^{-\lambda t}$

Let $f,g,\alpha:[0,\infty)\to\mathbb{R}$ be continuous nonnegative functions. Suppose $f$ is differentiable on $(0,\infty)$ and satisfies a differential inequality \begin{align} \frac{d}{dt}f(t)+g(t)\...
3
votes
0answers
15 views

Immersion $H^s(\Omega) \hookrightarrow H^{s'}(\Omega)$ but with fractional laplacian defined on $\mathbb{R}^N$?

I know that it holds this embeding $$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$ for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined ...
4
votes
1answer
66 views

Embedding Sobolev $H^1$ into $L^\infty$ space

My question is to eventually prove the following inequality: for $f\in H^1(\mathbb{R})=\{f,f'\in L^2\}$ $$\|f\|_{L^\infty}\leq a\|f\|_{L^2}+\frac{1}{a}\|f'\|_{L^2}, \forall a>0.$$ Here are my ...
0
votes
0answers
31 views

Minkowski’s integral inequality to triangle inequality

How do I get from the Minkowski's integral inequality to the triangle inequality (Minkowski's inequality)? I don't understand how exactly I get from the the Norm to the integral structure and back to ...
2
votes
1answer
16 views

Bound the value of function by integration of derivatives

Given $f(x,y) \in C^2([0,1]^2)$ (by which I mean $C^2$ in some open neighborhood), with $f_x, f_y, f_{xy} \in L^1([0,1]^2, dxdy)$ (which is sure case since they are continuous), does the following ...
1
vote
1answer
36 views

Find $I$ if $\int_0^1 \max\{f(a), f(x)\}dx\ge \max_{x \in I} f(x)$

Let $a\in [0,1]$. FInd all the closed intervals $I \subset [0,1]$ such that the following inequality holds for any convex differentiable function $f:[0,1]\to \mathbb{R}$ : $$\int_0^1 \max\{f(a), f(x)\}...
0
votes
1answer
69 views

Integral inequality: $3\left(\int_0^1 g(x)^2 dx\right)^3 \geq \int_0^1 g(x)^8 dx.$

Problem: Suppose that $g:[0,1]\to\mathbb{R}$ is a differentiable function such that $0\leq g'(x)\leq 1$ for each $x\in[0,1]$ and $g(0)=0.$ Prove that $$3\left(\int_0^1 g(x)^2 dx\right)^3 \geq \int_0^1 ...
0
votes
1answer
74 views

How to derive an inequality of 1-variable integration involving $f$

Let $f$ be a function defined on the interval $[a,b]$. Expand the inequality of double integral $\int_a^b\int_a^b[f(x)-f(y)]^2\,dA\geq\,0$ to derive an inequality of 1-variable integration involving ...
8
votes
1answer
152 views

$\int_0^\pi\left|\frac{\sin {nx}}{x}\right|dx\ge \frac{2}{\pi}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$

Question: Show that $$\int_0^\pi\left|\frac{\sin {nx}}{x}\right|dx\ge \frac{2}{\pi}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right).$$ My approach: We know that $$1+\frac{1}{2}+\cdots+\frac{1}{n}=\int_0^...
0
votes
0answers
27 views

Prove the inequality $ \left(\frac{\dot{z}^{T}(\xi) W z(\xi)}{\sqrt{z^{T}(\xi) W z(\xi)}}\right)^{2} \leq \dot{z}^{T}(\xi) W \dot{z}(\xi) $

Why does the following inequality hold: $$ \left(\frac{\dot{z}^{T}(\xi) W z(\xi)}{\sqrt{z^{T}(\xi) W z(\xi)}}\right)^{2} \leq \dot{z}^{T}(\xi) W \dot{z}(\xi) $$ It is used at the end of the ...
0
votes
1answer
35 views

Unable to derive an inequality related to integrals

I am unable to think on how to derive this inequality. Can someone please help!! Inequality is -> $\int_1^n logx dx$ $\leq$ log(n!) $\int_2^{n+1} logx dx $ . I tried using Integration by parts of ...
1
vote
1answer
68 views

Does the Cauchy-Schwarz integral inequality still hold for convergent improper integrals?

A few hours ago I posted this solution : https://math.stackexchange.com/a/3579886/629594 It seems to be all right at a first glance, given that I actually obtain the same constant as the other user, ...
14
votes
2answers
327 views

Prove that $\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)^{\!2}$

Prove that if $f:[0,1] \to \mathbb{R}$ is a continuously differentiable function with $\int_0^1 f(x)\,dx=0$, then $$\int_0^1 \big(1-x^2\big) \big(f'(x)\big)^2\,dx \ge 24 \left(\int_0^1 xf(x)\,dx\right)...
0
votes
1answer
53 views

Is $\frac{\phi(r)}{r}$ dominated by $\phi'(r)$?

Let $\phi:[0,1] \to \mathbb R$ be a smooth strictly increasing function satisfying $\phi(0)=0, \phi'(r)>0$ for every $r \in [0,1]$. Is it true that $\frac{\phi}{r} \le \phi'(r)$ for every $r \...

1
2 3 4 5
19