1
vote
3answers
65 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
39 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 ...
1
vote
2answers
37 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 ...
1
vote
0answers
60 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
votes
1answer
23 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} ...
7
votes
1answer
134 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!
3
votes
2answers
79 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 $$ ...
2
votes
1answer
68 views

Cauchy-Schwarz-like inequality of integrals

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$ I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$ so if we ...
2
votes
1answer
28 views

Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
0
votes
1answer
30 views

Some questions on the proof of Hoelders inequality.

I have some questions about the proof of Hoelder's inequality. Statement: Let $(X, \mathbb X, \mu)$ be a measure space. Let $p,q > 1$ with $1/p+1/q = 1$ and suppose that $f \in L_p(X)$ and $g \in ...
3
votes
2answers
131 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 $f>0$.
0
votes
0answers
51 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 ...
0
votes
1answer
55 views

Upsetting Integral inequality

I have two smooth scalar non-negative functions f,g in $\mathbb{R}^+$ such that $$ f(x) \leq g(x), ~\forall x\in \mathbb{R}^+ $$ which are integrable with finite integrals in $\mathbb{R}^+$. I would ...
3
votes
2answers
57 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...
4
votes
2answers
206 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
30 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: ...
8
votes
1answer
415 views

How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$

let $f(x)\ge 0,x\in [0,1]$, and is increasing in $[0,1]$ show that: There exists $g(x)\ge 0,x\in [0,1]$,and $g''(x)>0$, such $g(x)\le f(x)$, and such ...
0
votes
1answer
95 views

inequality with sup and Expectations

let $X_{t}$ Itô diffusion how we can get : $$ \sup_{x\in \mathbb{R}^{n}}|E^{x}[f(X_{t})]| \leq \sup_{y \in \mathbb{R}^{n}}|f(y)|\sup_{x\in\mathbb{R}}E^{x}[1]=\sup_{y\in\mathbb{R}^{n}}|f(y)|.$$ Here ...
2
votes
1answer
148 views

Prove that $\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$

Let $f$ be a continuously differentiable function on $[0,1]$ and $f(0)=0$. Prove that $$\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$$ Thank you!
1
vote
2answers
122 views

An integral inequality.

I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some $C>0$ where $a(t)>0$ and ...
2
votes
1answer
170 views

prove this $\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$

let $f\in C^1[0,2]$,and such $\int_{0}^{2}f(x)dx=0,f(0)=f(2)$, show that $$\int_{0}^{2}f^2(x)dx\le\int_{0}^{2}f'^2(x)dx$$ I think we must use $Cauchy$ inequality my idea:I have see this let ...
8
votes
3answers
379 views

Inductively prove that this sequence of integrals is bounded.

EDIT: I have an attempted solution to this in a post below, it is very long, but still incomplete. EDIT:Alright, I've pretty much almost finished my solution, but my biggest problem is the 2nd ...
8
votes
1answer
191 views

Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that $$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$ Prove that $$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$ Where should ...
14
votes
4answers
1k views

Prove $\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \, dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \, dx$

Prove that: $(1)$$$\int_0^{\infty } \frac{1}{\sqrt{6 x^3+6 x+9}} \ dx=\int_0^{\infty } \frac{1}{\sqrt{9 x^3+4 x+4}} \ dx$$ $(2)$$$\int_0^{\infty } \frac{1}{\sqrt{8 x^3+x+7}} \ dx>1$$ What I do for ...
9
votes
2answers
470 views

How prove this integral inequality $\int_{0}^{s}f(x)\,dx\le\int_{s}^{1}f(x)\,dx\le\dfrac{s}{1-s}\int_{0}^{s}f(x)\,dx$

let $f(x)>0$ is continuous and is increasing on $[0,1]$,and $s=\dfrac{\int_{0}^{1}xf(x)dx}{\int_{0}^{1}f(x)\,dx}$ show that ...
1
vote
1answer
60 views

The integral $\int_0^1\dfrac{(-x)^n}{1+x} dx $

How can I prove that: $\forall x \in \mathbb{N}\setminus {0} \quad \dfrac{-1}{n+1}\le \int_0^1\dfrac{(-x)^n}{1+x} dx \le \dfrac{1}{1+n}$ $\lim_{n\to+\infty}\Sigma_{i=1}^{n}\dfrac{(-1)^{i-1}}{i}$. ...
2
votes
1answer
134 views

How prove $\int_{0}^{1}F(x)dx\le \int_{0}^{a}G(x)dx$

if $G$ and $F$ are integrable,$a>0,G(x)\ge F(x)\ge 0$,and $$\int_{0}^{1}xF(x)dx=\int_{0}^{a}xG(x)dx$$ show that $$\int_{0}^{1}F(x)dx\le \int_{0}^{a}G(x)dx$$ ...
1
vote
2answers
253 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 ...
3
votes
2answers
272 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 ...
1
vote
2answers
68 views

Need to prove $\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$

I need to show that $$\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$$ I just know that if in $[a,b]$, $f(x)\le g(x)\le h(x)$, then ...
2
votes
2answers
132 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 ...
1
vote
0answers
89 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
2
votes
2answers
212 views

integration inequality [duplicate]

Possible Duplicate: Proving Integral Inequality Suppose $f(x)$ is differentiable on $[0,1]$ , $f(0)=0$ and $1\geq f'(x) >0 $ Prove that $\displaystyle\left(\int_{0}^{1} ...
16
votes
1answer
274 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: ...
17
votes
1answer
645 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
172 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 ...
9
votes
4answers
1k views

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 $g$ is ...
13
votes
2answers
518 views

Proof of bound on $\int t\,f(t)\ dt$ given well-behaved $f$

I got the following question by mail from someone I don't know from Adam. (Quoted in part.) if $f(t)$ continuously diff. on $[0,1]$ and a) $\int_0^1f(t)\ dt=0$ b) $m\le f\,'\le M$ on ...