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

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3
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29 views

Determining the best possible constant $k$, for an Integral Inequality

If $f : [0,\infty) \to [0,\infty)$ is an integrable function, then what is the best possible constant $k$, for which the following ineqality holds: $$\int_0^{\infty}f(x)dx \leq ...
3
votes
1answer
54 views

How to prove Godunova's inequality?

Let $\phi$ be a positive and convex function on $(0,\infty)$. Then $$\int_0^\infty \phi\left(\frac{1}{x}\int_0^x g(t)\,dt\right)\frac{dx}{x} \leq \int_0^\infty \phi(g(x))\frac{dx}{x}$$ The ...
7
votes
1answer
64 views

How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$

let $$f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$$ show that: $$e^x|f(x)|\le 2$$ My idea: let $$e^t=u$$ then $$|f(x)|=|\int_{e^x}^{e^{x+1}}\dfrac{1}{u}d\cos{u}|$$
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
2
votes
1answer
24 views

Help understanding an application of Jensen's inequality

This is from the book Pattern Recognition and Machine Learning by Christopher Bishop. The author states the following form of Jensen's inequality: $f\left(\int{xp(x)dx}\right) \leq \int{f(x)p(x)dx}$ ...
2
votes
1answer
38 views

How to prove this integral inequality?

Here is a problem: Let $B_r=\{ (x_1,x_2,\cdots,x_n)\in \mathbb{R}^n: x_1^2+x_2^2+\cdots+x_n^2<r^2\}.$ Let $f$ be a $C^1$ real function on $B_2$. Prove that $$\inf_{a\in R}\int_{B_2} ...
12
votes
2answers
141 views

$|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2}$

Let $f\in C^1([0,\pi],\mathbb R)$ such that $\displaystyle\int_0^\pi f(t) dt=0$ Prove that $\forall x\in [0,\pi],\displaystyle|f(x)|\leq \sqrt{\frac{\pi}{3}\int_0^\pi f'^2(t)dt}$ Failed ...
0
votes
0answers
60 views

Putnam 2013 B4 inequality

For any continuous real-valued function $f$ defined on the interval $[0,1],$ let $$\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|$$ Show that if $f$ ...
1
vote
4answers
77 views

An Integral Inequality Problem

How to establish the Integral Inequalities : $$ \displaystyle \int_0^1 \ln \sqrt{\dfrac{1-\cos x}{1+\sin x}} \,dx < \dfrac{1}{2}\ln 2$$ My attepmt : We have $\displaystyle $$(ii) \displaystyle ...
1
vote
1answer
33 views

Reference for $f \in L^{p,\infty} \cap L^{q}$ then $f \in L^r$ for $p < r \leq q$

Okay, so I think I've shown that if $f \in L^{p,\infty} \cap L^{q}$ with $p < q$ then $f \in L^r$ for $p < r \leq q$ where $L^{p, \infty}$ denotes the weak $L^p$ space. what I did was I wrote $$ ...
0
votes
3answers
52 views

How to prove this ${\pi\over4}\leq\int_0^{\pi\over2}e^{-\sin^2{x}}dx\leq{11\over32}\pi$

Can someone help to prove this? $${\pi\over4}\leq\int_0^{\pi\over2}e^{-\sin^2{x}}dx\leq{11\over32}\pi$$ I totally have no idea how to approach. Thanks.
0
votes
2answers
36 views

estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
3
votes
1answer
122 views

Integrals on a closed ball

Prove $$ \int_{\overline{B}(0,r)}c'D^{-1}c\ \exp(-1/2 y'D^{-1}y)\ dy >\int_{\overline{B}(0,r)}(y'D^{-1}c)^2\ \exp(-1/2 y'D^{-1}y)\ dy $$ where $ D\in\mathbb{R}^{n\times n} $ is a diagonal matrix, ...
1
vote
1answer
82 views

How prove this inequality $I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$

show that $$I=\int_{0}^{\sqrt{2\pi}}\sin{x^2}dx>0$$ follow is my methods: let $$x^2=t$$ then ...
2
votes
1answer
88 views

How can we prove this integral inequality ? $\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin{(2n+1)t}}{\sin{t}}\right|dt<\pi\left(1+\frac{\ln{n}}{2}\right)$

Use this $$\dfrac{1}{2}+\sum_{k=1}^{n}\cos{(kx)}=\dfrac{\sin{\left(n+\dfrac{1}{2}\right)x}}{2\sin{\dfrac{x}{2}}},x\neq 2m\pi,m\in\mathbb{Z}$$ to show that ...
2
votes
1answer
58 views

Upper bound of difference of squares of quantile standard normal

Let $\Phi$ denotes the cummulative standard normal distribution and $\Phi^{-1}$ denotes its inverse. Given $u,v\in[0,1)$. I'am going to find an upper bound of $$ ...
5
votes
0answers
38 views

If $f$ is $2 \pi$ periodic and $\int_{0}^{2 \pi} f(t) dt=0$ then $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$ [duplicate]

Given $f$ a real differentiable function, $2 \pi$ periodic such that $\int_{0}^{2 \pi} f(t) dt=0$ show that: $\int_{0}^{2 \pi} (f(t))^2 dt \le \int_{0}^{2 \pi} (f'(t))^2 dt$. When does equality hold? ...
0
votes
0answers
15 views

Finding conditions for joint probability density larger than the product of marginals

I was wondering if you could help me out. I have a joint probability distribution with density $f(x,y)$ and marginals $g(x)$ and $h(y)$ defined over the real line. Now, I would like to find a class of ...
0
votes
2answers
24 views

Inequalities giving incorrect solution

Question: Find the solution set for:$$\frac{|x|-1}{|x|-2} \geq 0$$ $x\not=\pm2$ My attempt: Let $|x| = y$, then inequality becomes $(y-1)(y-2)>=0$ Implies that: #1. $y-1\geq0$ and ...
2
votes
1answer
62 views

Integral inequality for nonnegative functions

I suppose that for $f(x) \geq 0$, $$ \left(\int_\Omega f\,dx\right)^2 \geq C\int_\Omega f^2\,dx $$ because $(a+b)^2 \geq a^2 + b^2$ for $a,b \geq 0$. Is this inequality true? How can I prove it?
0
votes
0answers
40 views

Reference for theorem? Inequality of integrals of increasing function over two distributions

I have a monotone increasing function $H(x)$ and two distributions with CDFs $F_1$ and $F_2$, where $F_1(x) \leq F_2(x)$ everywhere. The domain is $[0,\infty)$. This seems like it must be true: $$ ...
3
votes
2answers
99 views

How find the minimum of the value $k$ such this intergral $\int_{0}^{1}f^2(x)dx\le k\left(\int_{0}^{1}f(x)dx\right)^2$

Find the minimum of the value $k$, such have $$\int_{0}^{1}f^2(x)dx\le k\left(\int_{0}^{1}f(x)dx\right)^2$$ for any integrable function $f(x)$,and $1\le f(x)\le 2,x\in(0,1)$ My idea: ...
6
votes
3answers
113 views

How prove this inequality $\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}\sin^n{x}dx\ge 0$

show that: $$\int_{0}^{1}\sin{(x^n)}dx\ge\int_{0}^{1}\sin^n{x}dx\ge 0$$ My idea:maybe $\sin{(x^n)}\ge (\sin{x})^n?$ Thank you
5
votes
1answer
96 views

Integral inequality $\int_0^{+\infty}|\frac{\sin x}x|^p dx\leq\frac\pi{\sqrt{2p}}$

$p\geq2$, then we have $$\int_0^{+\infty}\Bigg|\frac{\sin x}x\Bigg|^p\,\mathrm dx\leq\frac\pi{\sqrt{2p}}$$ I try to use $\Bigg|\frac{\sin x}x\Bigg|\leq1$, and $\frac{\sin ...
2
votes
1answer
70 views

How does one prove or disprove this integral inequality for a $C^1([0,1])$ function with zero average?

By zero average, I mean $\int_0^1 f(x) dx = 0$. The inequality is $$ 2 \int_0^1 [f(x)]^2 dx \le \left(\int_0^1 |f(x)|dx\right)\left(\int_0^1 |f'(x)|dx\right). $$ Cauchy-Schwarz hasn't led me ...
3
votes
2answers
89 views

How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
1
vote
3answers
91 views

Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
0
votes
1answer
81 views

L1 convergence and Lp bounded implies Lq convergence

I have tried to solve this problem for almost a week and did not manage to, so I figured to ask it here: Let $(u_n)\to u$ in $L^1(0,1)$ strongly and let $\{u_n\}_{n\in\mathbb{N}}$ be bounded in ...
5
votes
1answer
91 views

How prove this integral inequality $6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx$

Let $f$ be a positive-valued,concave function on $[0,1]$,Prove that $$6\left(\int_{0}^{1}f(x)dx\right)^2\le 1+ 8\int_{0}^{1}f^3(x)dx$$ Let ...
1
vote
1answer
42 views

A problem about center of mass

Suppose $f(x)$ is positive, increasing and Riemann-integrable on the interval $[a,b]$. Let$$\bar{x}=\frac{\int_{a}^{b}{xf(x)\text{d}x}}{\int_{a}^{b}{f(x)\text{d}x}}.$$Prove ...
2
votes
0answers
38 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
6
votes
2answers
134 views

How prove this $\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$

Show that this intergral inequality $$\int_{0}^{2\pi}e^{\sin{x}}dx<2\pi e^{\frac{1}{4}}$$ I know this use Taylor's formula.But I think is very ugly,maybe this problem have simple methods.Thank you ...
1
vote
1answer
44 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
0
votes
1answer
39 views

hyperbolic inequality

Calculating some contour integral, I have to prove that $\int^{R+i}_{R}\frac{cosh(az)}{cosh(\pi z)}dz$ goes to zero if R goes to infinity. And we know that $\left|a\right|<\pi$. I want to use the ...
1
vote
2answers
41 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
0
votes
1answer
34 views

Somewhat L2 against H1 estimate; an inequality in H1

somehow I'm a little slow on this one: Let $\Omega = [0,1]^2 \subseteq \mathbb{R}^2$ and $\emptyset \neq D \subsetneq \Omega$. Do constants $c_1,c_2,c_2\in\mathbb{R}_{\geq 0}$ exist such that $$ c_1 ...
1
vote
0answers
38 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
4
votes
1answer
110 views

Integral inequality with a function twice differentiable

Let $f:[0,1]\longrightarrow\mathbb{R}$ be a function twice differentiable with continous second derivative and $f(1)=f(0)$. The inequality: $$\int_{0}^{1}(f''(x))^2dx\geq ...
0
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0answers
42 views

Median-median inequality

An elementary result from Chebyshev's theorem is that the median and mean of a random variable do not differ by more than one standard deviation. I'm curious if there is a similar result for ...
0
votes
1answer
26 views

An integral inequality with little information

$u,v$ are scalar fields on $V\subset\mathbb{R}^3$ such that $\nabla^2 u=0$ on $V$ and $u=v$ on $\partial V$. Prove that: $$\int_V|\boldsymbol{\nabla} ...
0
votes
0answers
30 views

Prove Inequality with Weight Function

The following question concerns a proof from Gilbarg and Trudinger's book titled "Elliptic Partial Differential Equations of Second Order", 2001, p.223 Without giving all the details of the proof in ...
7
votes
1answer
169 views

Integral Inequality $|\int_0^x f(t)dt|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|$

Let $f\in C^2(\mathbb{R})$ such that $f(1)=\int_0^1f(x)dx=0$. Prove that $$\left|\int_0^x f(t)dt\right|\le \frac{2}{81}\max_{0\le x\le1}|f^{''}(x)|\, \forall x\in [0,1].$$ Thanks in advance!
1
vote
0answers
49 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
4
votes
1answer
93 views

Integral of composition [duplicate]

Prove that if $f,g:[0,1]\rightarrow[0,1]$ - continuous functions and f is strictly increasing then $$\int\limits_0^1f(g(x))dx\leq\int\limits_0^1f(x)dx+\int\limits_0^1g(x)dx.$$ I tried to prove that ...
3
votes
2answers
91 views

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 $$ ...
0
votes
1answer
58 views

How to apply Plancherel Theorem here?

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R) and the L^1 norm of f(x) is less than or equal to 8 times (the L^2 norm of f(x)) times the L^2 norm ...
1
vote
1answer
47 views

use plancherel theorem to prove an integral inequality

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R). I'm sorry I don't know how to use Latex to post the problem. The origional problem is here: ...
1
vote
0answers
38 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
0
votes
1answer
141 views

Cauchy–Schwarz inequality on vector-valued L2 space

Let $f$ and $g$ be square-integrable, $\mathbb{R}^n$-valued functions, i.e., $$ \| f \|_2^2 = \int \|f(t)\|^2 dt < \infty $$ where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$. I am looking ...
3
votes
4answers
144 views

Integral inequality for continuous function

Let $ f $ be a continuous, real-valued function on $[0, 1] $. Show that $$\int_0^1 \int_0^1 |f (x)+f (y)| dx dy \ge \int_0^1 |f (x)| dx $$ I tried to dissect the square in triangles and use ...