8
votes
2answers
174 views

Integral inequality: $\def\intd{\,\mathrm d}\int_a^b(f'(x))^2\intd x-2\big(f(a)+f(b)\big)^2\geq\frac8{(b-a)^2}\int_a^b(f(x))^2\intd x$

I have a problem which I think is wrong. Let $f: [a,b] \to \mathbb{R}$ be a differentiable function with $f'$ continuous such that $$\int_a^b f(x) \intd x = f\left(\frac{a+b}{2}\right) = 0$$ ...
1
vote
1answer
30 views

Lower bound on $F$ under the assumption $\theta F(s)\le sF'(s)$

Let $F(s)=\displaystyle \int_0^{s}f(t)\,\mathrm dt$. We suppose that there exists $\theta>2$ such that $\theta F(s)\le f(s)s$ for all $s\in \mathbb{R}$ and that $F(s)>0$ for all ...
5
votes
1answer
93 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
61 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 ...
0
votes
0answers
27 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
41 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
165 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
2answers
37 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 ...
2
votes
1answer
94 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
61 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 $$ ...
2
votes
1answer
63 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?
5
votes
1answer
108 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
77 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 ...
0
votes
1answer
104 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 ...
4
votes
1answer
105 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 ...
0
votes
1answer
60 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
51 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: ...
0
votes
1answer
158 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
149 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 ...
11
votes
6answers
398 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
4
votes
2answers
161 views

Integral inequality $\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$

How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
1
vote
1answer
75 views

Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \begin{equation} \int_{a}^{b} ...
8
votes
1answer
138 views

A continuous function integral inequality

Let $m$ be a positive integer. $f\colon[0,\infty)\to[0,\infty)$ is a continuous function such that $f(f(x))=x^m,\forall x\in[0,\infty)$. Show that $$\int_0^1f^2(x)\,dx\ge\frac{2m-1}{m^2+6m-3}$$
0
votes
2answers
79 views

An Hardy-Littlewood like inequality

Let $\mu$ be a finite Borel measure on $R$. Show that $|\{x\in R : \sup_{r>0} \frac{1}{2r} \mu ([x-r,x+r]) \ge \lambda \}| \le \frac{C}{\lambda} \mu(R)$ for any $\lambda > 0$ and some absolute ...
6
votes
2answers
288 views

Prove that if f(x) is integrable, then so is e^(f(x)).

So here is my question: I'm working on a homework problem that deals with Jensen's Inequality. It is a rather simple application, I believe, but I'm a little stuck. Here is the problem, along with ...
0
votes
0answers
53 views

inequality of integrals

Let $f$ be of bounded variation on $[0,1]$ and $g:[0,1]\rightarrow \mathbb{R}$ be Lebesgue integrable on $[0,1]$. Prove \begin{equation} |\int_0^1fg d\lambda|\leq (|f(0)|+\text{Var}_{[0,1]}f)\cdot ...
2
votes
1answer
27 views

An inequality in $L^p$-spaces

Let $\{f_k\}_{k=1}^{\infty}$ be a sequence in $L^p(\Omega,\Sigma,\mu)$ for $1\leq p<\infty$. Suppose $0<c=\inf_k \lVert f_k\rVert_p\leq \sup_k \lVert f_k\rVert_p=C<\infty$ and $f_if_j=0$ for ...
4
votes
2answers
229 views

How prove this integral inequality $2\int_{-1}^{1}f(x)g(x)dx\ge\int_{-1}^{1}f(x)dx\int_{-1}^{1}g(x)dx$

Let $f:[-1,1]\longrightarrow \mathbb{R}$ be increasing on $[0,1]$ and even, i.e. $f(x)=f(-x)$ $\forall x\in [-1,1]$. Let $g:[-1,1]\longrightarrow \mathbb{R}$ be convex, i.e. $g(tx+(1-t)y)\le ...
0
votes
0answers
31 views

How to prove a duality of $L^p$ spaces? [duplicate]

Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $f:\Omega\longrightarrow \mathbb{R}$ be a measuable function. Let $1\leq p< \infty$ and $1/p+1/q=1$. Prove that the following are equivalent: ...
2
votes
2answers
92 views

an amazing inequality of $L^p$ spaces, how to prove it?

Let $\Omega_1,\Sigma_1,\mu_1$ and $\Omega_2,\Sigma_2,\mu_2$ be $\sigma$-finite complete measure space and $f:\Omega_1\times\Omega_2\longrightarrow [0,+\infty)$ be $\mu_1\otimes\mu_2$ measurable. Then ...
0
votes
2answers
50 views

Prove that $\left \|x(t_0) \right \|\exp \left (-\int_{t_0}^{t} \left \|A(t_1) \right \|\mathrm{d}t_1 \right )\le \left \| x(t) \right \|$

I have a problem: For $\dfrac{dx}{dt}=A(t)x$, where $A(t)\in C\left [t_0,+\infty \right )$. Prove that: $$\left \|x(t_0) \right \|\exp \left (-\int_{t_0}^{t} \left \|A(t_1) \right \|\mathrm{d}t_1 ...
1
vote
1answer
106 views

If the weighted Lp norm of a measurable function is finite, is the weighted Lp norm of the antiderivative also finite?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} dx < \infty. $$ Define $g : \mathbb{R} \rightarrow \mathbb{R}$ to be $$ ...
2
votes
1answer
233 views

Simplification of integral with division between summations

Considering that $$\sum_{j = 0}^{\infty} \int f_j(x) < \infty$$ and $$\sum_{j = 0}^{\infty} \int g_j(x) < \infty$$, $\forall x \in \mathrm{R} : f(x) \gt 0, g(x) \gt 0$. How can I simplify the ...
3
votes
1answer
247 views

integral inequality with derivative

Let a function $f:[0,1]→\mathbb{R}$ have a continuous derivative and $$\int_{0}^{1}f(x)dx=0$$ Show that for every $\alpha \in [0,1]$, $$\left|\int_{0}^{\alpha}f(x)dx\right|\le\frac{1}{8} \mathbb{sup} ...
1
vote
1answer
83 views

Exercice on periodic function

Let $f$ be a periodic function, $\mathcal{C}^1$ on $\mathbb{R}$ such that: $$\displaystyle\int_0^{2 \pi} f(t) \, dt = 0$$ $$f(2 \pi) = f(0)$$ Prove that $$\forall t \in [0,2 \pi]: \int_0^{2 \pi} ...
2
votes
1answer
70 views

Inequality in Sobolev Space

Given $\Omega \subset \mathbb{R}^3$, prove $\forall u, v, w \in H^{1,2} (\Omega)$ it holds that $ | \int_{\Omega} u \frac{\partial v}{ \partial x} w dx | \leq \| u \|_{1,2,\Omega}\|v \|_{1,2,\Omega}\| ...
3
votes
1answer
177 views

Hilbert's Inequality

Could you help me to show the following: The operator $$ T(f)(x) = \int _0^\infty \frac{f(y)}{x+y}dy $$ satisfies $$\Vert T(f)\Vert_p \leq C_p \Vert f\Vert_p $$ for $1 <p< \infty$ where ...
3
votes
2answers
145 views

An integral inequality related to Taylor expansion

Problem. Let $f:[a,b]\to\mathbb{R}$ be a function such that $ f\in C^3([a,b])$ and $f(a)=f(b)$. Prove that $$ ...
1
vote
2answers
271 views

Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$

Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$ My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...
2
votes
1answer
117 views

integer Random Walk with step size governed by a distribution.

This problem is for a final exam I am taking in a graduate probability class. Collaboration has been explicitly allowed, but with the remark that the professor felt he couldn't stop us even if he ...
4
votes
2answers
284 views

$3\int_{0}^{1}(f'(x))^2dx \geq (2\int_{0}^{1}f(x)dx)^2 \impliedby 2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$

Let $f : \mathbb{R} \to \mathbb{R} $ be a differentiable function. Suppose that $2\int_{0}^{\frac{1}{2}}f(x)\,\mathrm dx=\int_{\frac{1}{2}}^{1}f(x) \,\mathrm dx$ Show that $$3\int_{0}^{1}(f'(x))^2 ...
8
votes
1answer
236 views

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 ...
2
votes
2answers
135 views

Given $\int_0^1 x f(x)dx=0$, show that $\int_0^1|1-f(x)|dx>1/2$

I have seen this statement before, and I would like to use it in a proof I am working on. I do not quite remember the condition on $f$--whether it is just integrable or continuous. Can someone point ...
16
votes
1answer
294 views

Prove the following integral inequality

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: ...
18
votes
1answer
654 views

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 ...
3
votes
1answer
182 views

Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$.

Suppose $f(x)$ and $g(x)$ are positive measurable functions defined on $(0,1)$, satisfying $f(x)g(x)\ge1$ for any $x\in(0,1)$. Prove that $\int_0^1f(x)dx$$\int_0^1g(x)dx\ge1$. Totally no idea about ...
2
votes
2answers
63 views

How to prove the first inequality?

Assume $\mu(X)=1$ and $h\ge 0$ is measurable, if $A=\int_{X}hd\mu$, prove that $$\sqrt{1+A^{2}}\le \int_{X}\sqrt{1+h^{2}}d\mu\le 1+A$$ I am wondering how to prove the first part of the ...
7
votes
2answers
465 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
2
votes
2answers
590 views

Proof of Clarkson's Inequality

Trying to find a proof for Clarkson's inequality, which states that if $2 \leq p < \infty$, then for any $f, g \in L^p$, we have that $$\left|\left|f+g\right|\right|_p^p + ...
1
vote
1answer
145 views

If $\int_x^1f(t)dt\ge\frac{1-x^2}2$, $x\in[0,1]$, prove that $\int_0^1f(t)^2dt\ge1/3$.

Let $f$ be continuous on $[0,1]$ that satisfies $\int_x^1f(t)dt\ge\frac{1-x^2}2,x\in[0,1]$. Prove that $\int_0^1f(t)^2dt\ge1/3$.