For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
Questions tagged [integral-inequality]
1106 questions
3
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How to prove $|xe^{-x^2}\int_{0}^{x}e^{t^2}dt-ye^{-y^2}\int_{0}^{y}e^{t^2}dt|<|x-y|$ for $x\ne y$
Let $x,y$ be distinct positive numbers. Show that
$$\left|xe^{-x^2}\int_{0}^{x}e^{t^2}dt-ye^{-y^2}\int_{0}^{y}e^{t^2}dt\right|<|x-y|.$$
I found this problem when I was dealing with the problem below, but I can't solve it. Thank you.
Show…
math110
- 93,304
3
votes
1 answer
How prove this integral inequality$\left(\int_{0}^{+\infty}\left(\frac{1}{x}\int_{0}^{x}|f(t)|dt\right)^pdx\right)^{\frac{1}{p}}$?
let $p>1$, and $f$ is continuous and $\displaystyle\int_{0}^{+\infty}|f(t)|^p|dt$ is convergence,show…
user94270
3
votes
1 answer
Integral inequality
Let $f$ be a continuously differentiable real-valued function on $[0,b]$, where $b>0$, with $f(0)=0$. Prove that
$$\int\limits_0^b\frac{f(x)^2}{x^2}dx\leq4\int\limits_0^b f'(x)^2dx.$$
Thank you!
Anton
- 75
3
votes
5 answers
Prove with integration the inequality $e(\frac{n}{e})^n < n! < n \times e(\frac{n}{e})^n$
Prove with integration the inequality, I need some advice about how to start prove it.
I know that if function is Monotonically increasing function so :
$$ f(1)+\int^n_1f(x)dx\leq f(1)+f(2)+....+f(n)\leq f(n)+\int^n_1f(x)dx$$
$$e(\frac{n}{e})^n <…
Ofir Attia
- 3,136
2
votes
1 answer
A Expectation question
I'm not sure if the following statement is right or not. But it looks right to me though I don't know how to prove it. Can someone help?
Suppose $X$ and $Y$ is independent and $EX=EY=0$, do we always have $E(|X+Y|)=E(|X-Y|)$ then? If so, how to…
user84325
- 35
2
votes
1 answer
Lyapunov inequality for BVPs
In some papers, it is said that Lyapunov [1, p. 406] proved the following result.
Let $p:[a,b]\to\mathbb{R}$ be a continuous nonnegative function.
If the…
bkarpuz
- 833
2
votes
1 answer
$x+y+z+4xyz=2$ prove that$ xy+yz+xz\le 1$
Let $x$,$y$,$z$ be non-negative real numbers, for which:
$x + y + z + 4xyz=2$
Prove that:
$xy + yz + zx \le1$.
I am sure this can be done with Cauchy-Schwarz or AM-GM, but I have long forgotten how to use those...Any help is appreciated !
Sartr
- 179
2
votes
0 answers
integral inequality involving $\sup|f'|$
Let $f:[0,1]\rightarrow \mathbb R$ be continuous function differentiable on $(0,1)$ with property that there exists $a \in (0,1]$ such that
$$\int_{0}^a f(x)dx=0$$
Prove that
$$\left|\int_{0}^1 f(x)dx \right|\le \dfrac {1-a} 2 \cdot \sup_{x\in…
Booldy
- 1,070
1
vote
2 answers
Integral mean inequalities
If $f \in C[0,1]$, then should be true that
$$\left( \int |f|^p\right)^{1/p} \leq \left( \int |f|^q\right)^{1/q}$$
for $1
evgeny
- 3,781
1
vote
0 answers
inequality question with integrals
There is a question that, I think, has a definite answer, but I can't figure it out. Given are three real valued functions, $f$,$g$, $w$, of a real variable $x$. The functions are non-negative, i.e., $f(x) \ge 0$, $g(x) \ge 0$ and $w(x) \ge 0$ for…
Physicist
- 171
1
vote
2 answers
How prove this integral inequality $\min_{x\in[0,1]}f(x)\ge-\int_{0}^{1}|f'(x)|dx$
let $f(x)$ can change sign in $x\in [0,1]$ and is continuous derivative function, show that
$$\min_{x\in[0,1]}f(x)\ge-\int_{0}^{1}|f'(x)|dx$$
My try:since $f(x)$ change sign in $x\in [0,1]$,
then there exsit $\xi\in [0,1]$ such $f(\xi)=0$.
then…
math110
- 93,304
1
vote
0 answers
Integral Gronwall inequality with negative coefficient
In Wikipedia https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality#Integral_form_for_continuous_functions we can find the following statement:
Let $I$ denote an interval of the real line of the form $[a, ∞)$ or $[a, b]$ or $[a, b)$ with $a <…
xiaozang
- 11
1
vote
0 answers
Question on integral inequality
I am reading the following proof of integral inequality lemma:
Let $b(t)$ and $f(t)$ be continuous functions for $t \geq \alpha$, and $v(t)$ is a differentiable function for $t \geq \alpha$. Suppose that
\begin{equation}
v'(t) = b(t)v(t) + f(t),…
Abel
- 266
1
vote
0 answers
A integral inequality in the paper "The concentration-compactness principle".
In the paper by P. L. Lions, I am confused with one inequality (Page 125, Line 8 from below.) as the following.
\begin{align}
&…
Shuai Yuan
- 11
- 3
1
vote
2 answers
Inequality involving the definite integrals of $\cos(x) f(x)$ and $\sin(x) f(x)$
This is a reposting of a previous question whose answer was accepted, but after having introduced a crucial difference in the hypothesis which would now require a different answer.
Considering this trivial (at $a=\pi$) definite integrals…
Luca
- 182