Questions tagged [integral-inequality]
By Michael Henderson •
Ask Question
For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
1,018 questions- Bountied 0
- Unanswered
- Frequent
- Score
- Unanswered (my tags)
A simple special case of Gronwall's inequality for Dini derivatives
Let $I=[t_0,t_1)\subset \Bbb R$ an interval and $a,b,c\ge0$ with $a>c$. Assume that $f\colon I\to\Bbb R$ is a continuous function with $$\tag{1} f(t)-f(s)\le \int_s^t\left( -af(r)+be^{-cr} \right) ... real-analysis ordinary-differential-equations integral-inequality dini-derivative- 3,971
Functional inequality with integral function [closed]
Given a function $f:[0;1]\to[0;1]$ such that $f(x)\leq2\int_0^x f(t)dt$, prove that $f(x)=0$ $ \forall x\in [0;1]$. I've observed that the function has to be concave down in his domain and that $... calculus integral-inequality functional-inequalities- 33
Show that $\int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4$
I couldn't prove this inequality $$ \int_2^{+ \infty} \frac{\log(t)^2}{t(t-1)}dt \leq 4 $$ I've tried integration by parts but it doesn't work. calculus integration inequality integral-inequality- 55
What is the correct version of the Gronwall lemma? Can the sign of u(t) be variable?
In the various forms of the Grönwall lemma in integral form are stated for NON NEGATIVE function $\phi$ And this coincides with what is written my ... real-analysis ordinary-differential-equations integral-inequality- 1,136
Integration over the intersection of a hypersphere and half spaces
For $x\in\mathbb{R}^n$, $u\in\mathbb{R}^n$ and $0<c<\Vert u\Vert$, how could I compute (or find the upper bound of) the following integral $$\int_{x^\top x =1,~\vert u^\top x\vert\leq c}1dx.$$ ... integration inequality integral-inequality- 53
Derivation of Hölder Inequality through Young's Inequaliy
I am having trouble following a proof where Young's Inequality is being used to derive Hölder's Inequality. More precisely, there is a particular and final step that utilizes integration in order to ... measure-theory lp-spaces integral-inequality holder-inequality young-inequality- 67
How to evaluate the following integral might be related to the modified Bessel Function of first kind?
Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation. $$\mathbb{E}_{Z_{1},Z_{2},\... definite-integrals expected-value bessel-functions integral-inequality jensen-inequality- 21
How to solve the following double integral with the Bessel function?
Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation. $$\mathbb{E}_{Z_{1},Z_{2}} \... probability-distributions definite-integrals bessel-functions integral-inequality gamma-distribution- 21
A Reverse Cauchy-Schwarz Inequality.
My question is closely related to the question here, where it was established that if $f:[a,b] \longrightarrow \mathbb{R}$ such that $f \in L^2([a,b])$ and there exists $M=M_f,m=m_f>0$ so that $m_f ... real-analysis inequality lp-spaces cauchy-schwarz-inequality integral-inequality- 1,427
Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$
The number of continuous functions $f:\left[0,\frac{3}{2}\right]\rightarrow (0,\infty)$ satisfying the equation$$4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$$ ... inequality cauchy-schwarz-inequality a.m.-g.m.-inequality integral-inequality- 7,092
Showing inequality for the norm of an integral equation
Let $\xi \in \mathbb{R}^2$, $\Phi \in C^0([0, +\infty[, \mathbb{R}^{2x2})$ a bounded function. Let $y:[0, +\infty[ \rightarrow \mathbb{R}^2$ a solution of the following integral equation, $$ y(x) = e^{... ordinary-differential-equations inequality integral-inequality integral-equations- 577
Is this kind of inverse substitution justified?
I'm trying to prove that for every integer $n\geq 0$ we have $$\int_0^{\pi/2} (1+\cos t)^ndt\geq \frac{2^{n+1}-1}{n+1}.$$ I started out by rewriting the RHS as $\int_0^{1}(1+x)^ndx$ and substituting $... real-analysis calculus integration inequality integral-inequality- 41
Inequality involving moments of a distribution [closed]
Let X be a real random variable. Under what conditions on the distribution do we have that $$\mathbb{E}( X^{2n + 2}) \geq \mathbb{E}( X^{2n}) \mathbb{E}( X^{2})$$ for all integer $n$? I tried using ... inequality expected-value cauchy-schwarz-inequality integral-inequality- 19
What functions satisfy $\int_a^b f(x) c(x) \, dx \ge 0$ for all convex functions $f$?
This is an attempt to generalize Prove that $\int_{0}^{2\pi}f(x)\cos(kx)dx \geq 0$ for every $k \geq 1$ given that $f$ is convex. Inspired by that question and the given answers, I have the ... real-analysis convex-analysis integral-inequality- 87.3k
an integral inequality from "Inequality" by Hardy,Littlewood and Polya Chapter7
If $f\in C^1[0,1)$ and $f(0)=0$ , show that $\int_0^1\frac{|f(x)|^2}{x^2}dx\leq4\int_0^1|f^{'}(x)|^2dx$ The book solves the problem using variational method. But I want to seek for an elementary ... integration integral-inequality- 23
15 30 50 per page12345…68 Next
