For questions inequalities which involves integrals, like Cauchy-Bunyakovsky-Schwarz or Hölder's inequality. To be used with (inequality) tag.
Questions tagged [integral-inequality]
1157 questions
114
votes
1 answer
Is this continuous analogue to the AM–GM inequality true?
First let us remind ourselves of the statement of the AM–GM inequality:
Theorem: (AM–GM Inequality) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have $$\frac1N\sum_k x_k \geqslant \left(\prod_k x_k\right)^{\frac1N}$$
It…
user1892304
- 2,894
46
votes
1 answer
Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$
Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$
How would we prove this? Does this follow from Cauchy Schwarz? Intuitively this is how I see it: In the LHS we could have a…
fourierguy
- 463
40
votes
3 answers
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:
$$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$
Larry Eppes
- 1,009
38
votes
2 answers
On the inequality $ \int_{-\infty}^{+\infty}\frac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \le n^{3/2}\pi.$
$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real
roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$
It's easy to see that the degree of $ p$ has to be even.
For…
jack
- 1,266
28
votes
1 answer
Geometric interpretation of Hölder's inequality
Is there a geometric intuition for Hölder's inequality?
I am referring to $||fg||_1 \le ||f||_p ||f||_q $, when $\frac{1}{p}+\frac{1}{q}=1$.
For $p=q=2$ this is just the Cauchy-Schwarz inequality, for which I have geometric intution: The projection…
Asaf Shachar
- 25,967
27
votes
4 answers
Prove that $\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1$.
Let $f\in C^{3}[0,1]$such that $f(0)=f'(0)=f(1)=0$ and $\big|f''' (x)\big|\le 1$.Prove that $$\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1 .$$
I couldn't make much progress on this problem. I thought that maybe I should try using polynomial…
JustAnAmateur
- 853
27
votes
5 answers
Jensen's inequality for integrals
What nice ways do you know in order to prove Jensen's inequality for integrals? I'm looking for some various approaching ways.
Supposing that $\varphi$ is a convex function on the real line and $f$ is an integrable real-valued function we have…
user 1591719
- 44,987
23
votes
1 answer
Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$
Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $f(f(x)) = x^2, \forall x \in [0,\infty)$. Prove that $\displaystyle{\int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}}$.
All I know about this function is that $f$ is bijective, it is…
C_M
- 3,749
23
votes
2 answers
A tricky integral inequality
A friend has submitted this problem to me:
Let $0
Gabriel Romon
- 36,881
22
votes
3 answers
Reverse Cauchy Schwarz for integrals
Let $f,g$ be two continuous positive functions over $[a,b]$
Let $m_1$ and $M_1$ be the minimum and maximum of $f$
Let $m_2$ and $M_2$ be the minimum and maximum of $g$
Prove that $$\sqrt{\int_a^bf^2 \int_a^b g^2}\leq…
Gabriel Romon
- 36,881
22
votes
2 answers
Proof of Wirtinger inequality
Quoting from Ana Cannas da Silva's book on Symplectic Geometry:
"As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have
$$
\int_a^b\Big|\frac{\mathrm{d}f}{\mathrm{d}t}\Big|^2\mathrm{d}t…
Brightsun
- 6,963
21
votes
1 answer
Do inequalities that hold for infinite sums hold for integrals too?
Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$ .
Let there be a function…
Amr
- 20,470
17
votes
6 answers
How prove this $\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx$
let $f\in C^{(1)}[a,b]$,and such that $f(a)=f(b)=0$, show that
$$\int_{a}^{b}f^2(x)dx\le (b-a)^2\int_{a}^{b}[f'(x)]^2dx\cdots\cdots (1)$$
My try: use Cauchy-Schwarz inequality
we have
$$\int_{a}^{b}[f'(x)]^2dx\int_{a}^{b}x^2dx\ge…
math110
- 94,932
- 17
- 148
- 519
17
votes
1 answer
Holder's inequality for infinite products
In analysis, Holder's inequality says that if we have a sequence $p_1, p_2, \ldots, p_n$ of real numbers in $[1,\infty]$ such that $\sum_{i=1}^n \frac{1}{p_i} = \frac{1}{r}$, and a sequence of measurable functions $f_1, f_2, \ldots, f_n$, then…
JHF
- 11,714
17
votes
2 answers
An integral inequality (one variable)
Anyone has an idea to prove the following inequality?
Let $g:\left(0,1\right)\rightarrow\mathbb{R}$ be twice differentiable
and $r\in\left(0,1\right)$ such…
Binjiu
- 731