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

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one inequality involving two stochastic processes

I am having trouble proving one inequality involving two stochastic processes. The problem seems simple but I just cannot handle it. Any help would be appreciated. $S_t$ and $C_t$ are two positive ...
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0answers
22 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
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122 views

Prove $\int_{0}^{1} |\frac {f^{''}(x)}{f(x)}| dx \geq 4$ [on hold]

I find an interesting theorem,but have no idea to prove it. $f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $ Prove that if $\int_{0}^{1} \bigl|\frac ...
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0answers
37 views

Hölder's inequality and log convexity of $L^{p}$ norm

Hölder's inequality of $L^{p}(X,\mu)$ $\left\Vert fg \right\Vert_{r} \leq \left\Vert f \right\Vert_{p} \left\Vert g \right\Vert_{q}$ where $0<p,q,r\leq \infty$ and ...
3
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1answer
44 views

on an exponential inequality

I'm working on proving the following inequality:$$((k/2)!)^2k^{-k}\ge(2e)^{-k}$$ for any positive even integer $k$. I can use Stirling formula to prove it for large $k$'s, but I want a proof that ...
2
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1answer
141 views

Reference or proof for an integral inequality

The following seems believable and quasi-intuitive to me, but it also doesn't quite seem trivial, and I'm not sure whether I've seen it stated before. Let $f$ be a complex-valued integrable function ...
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1answer
23 views

how to prove this equality $||f||_{L^p}^{p}=p\int_0^{+\infty} \lambda^{p -1}\mu(E^f_\lambda) d\lambda$

Let $(X,B(X),\mu)$ be a measure space, suppose there is a function f that is measurable Define the distribution function ${\mu(E_\lambda^f): {\mathbb R}^+ \rightarrow [0,+\infty]}$ How to prove ...
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2answers
43 views

Minkowski inequality of infinite sum

For $1\leq p <\infty,$ Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$ Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty ...
2
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1answer
37 views

Cauchy Schwartz with integrals of integrable functions

I was reading and doing problems from Spivak's Calculus on Manifolds. Q1-6 (a) stumped me a little. Let $f$, $g$ be integrable on $[a,b]$. Prove that $$\left| \int_a^b f\cdot g \; \right | \leq ...
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0answers
21 views

Detail in a proof about energy minimizing harmonic maps

Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional ...
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1answer
43 views

What inequality was applied?

I'm reading some probabilistic paper and have a trouble with understanding some part. Here is this part: mu is a Lipschitz function and M is the Lipschitz constant, and: What inequality was ...
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2answers
45 views

Inequalities for combinations of $\int f $ and $\int (1/f)$ where $m\le f\le M$ on an interval

Let $f\in C[a,b]$. Assume that $\min_{[a,b]}f=m>0$ and $M=\max_{[a,b]}f$. Which one is true? a. $$\frac{1}{M}\int_a^bf(x)dx+m\int_a^b\frac{1}{f(x)}dx\geq 2\sqrt{\frac{m}{M}}(b-a)$$ b. ...
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3answers
54 views

How to prove Cauchy-Schwarz integral inequality?

The Cauchy-Schwarz integral inequality is as follows: $$ \displaystyle \left({\int_a^b f \left({t}\right) g \left({t}\right) \ \mathrm d t}\right)^2 \le \int_a^b \left({f \left({t}\right)}\right)^2 ...
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0answers
19 views

An exponential integral inequality [duplicate]

Let $f:[0,1]\to \mathbb{R}^+$ be a continuous function. Prove that $$ \int_{0}^{1}f(x)\,\mathrm{d}x-\exp\left(\int_{0}^{1}\log f(x)\,\mathrm{d}x\right)\leq \max_{0\le x,y\le ...
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0answers
87 views

How to show $\int_0^1 f^3 dx < ( \int_0^1 f dx )^2 $ [duplicate]

Assume $f$ is $C^1([0,1])$ and $f(0)=0$ and $ 0 < f' \le 1 $ then I want to show that $$\int_{0}^{1} f^3 dx < \left( \int_{0}^{1} f dx \right)^2 $$ my tries : I want to use a similar way like ...
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2answers
233 views

How prove this integral inequality $4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$

Question: let $$f(0)=0,f(1)=1, f''(x)>0,x\in (0,1)$$ let $k>2$ are real numbers,show that $$4(k+1)\int_{0}^{1}(f(x))^kdx\le 1+3k\int_{0}^{1}(f(x))^{k-1}dx$$ This problem is from china ...
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1answer
35 views

Does this variation of Jensen's inequality hold?

The original Jensen's inequality in probability theory is generally stated in the following form: if $X$ is a random variable and $f$ is a convex function, then $f \left(\mathbb{E}[X]\right) \leq ...
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0answers
35 views

Prove that $\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}$ [duplicate]

Let $(X,\mathcal{A}, \mu)$ be any measure space and let $u \in \bigcap_{p\in [1,\infty]} \mathcal{L}^p(\mu)$. Then $$\lim_{p\to \infty} \|u\|_p = \|u\|_{\infty}.$$ I have already proved the ...
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1answer
36 views

Generalised Holder ineq

Prove the following generalisation of Holder's inequality $$\int | u_1 \cdot ... \cdot u_N | d\mu \leq \|u_1\|_{p_1} \cdot ... \cdot \|u_N\|_{p_N}$$ for all $p_j \in (1,\infty)$ such that ...
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1answer
49 views

$L^1$ and $L^2$ norm inequaliy

Consider real valued function $f$ defined on $[0, T]$. L1 norm and L2 norm of function $f$ are given by $$ \|f\|_1=\int_0^T |f(t)| \, dt $$ and $$ \|f\|_2=\sqrt{\int_0^T |f(t)|^2 \, dt } $$ Then we ...
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2answers
69 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
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0answers
113 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
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218 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
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1answer
108 views

How to prove this integral-inequality.

Suppose $f$ is twice differentiable and satisfies $f(0)=0$. Prove the inequality. $$\int_0^1 |f(x)f'(x)| dx \le\ \frac{1}{2} \int_0^1 |f'(x)|^2 dx $$ This is a problem from undergraduate math ...
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23 views

Sufficient conditions for inequalities

Assume $f$ and $g$ are two Lebesgue integral functions on interval $[a,b]$. Do we have any nontrivial sufficient conditions on $f$ and $g$ such that $$\forall a\leq x<y\leq b,\quad ...
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1answer
28 views

Existance of solution of $Ax < b$

How to check if the inequality $Ax < b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$
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1answer
37 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
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24 views

Prove an integral inequality [closed]

It is true that $$ \int \int\int f(y) f(z) g(x-y)g(x-z)g(x) dx dy dz \leq || f||_2^2 ||g||_2^2 $$
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1answer
35 views

The best constant in an integral inequality

I find a interesting inequality. Suppose that $y=y(x)$ is a differentiable function in $(0,L)$ and $y(0)=y(a)=0$. Consider the fraction $$ F[y]=\frac{\int_0^{L}\vert y'\vert^2dx}{\int_0^L\vert ...
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1answer
51 views

Proving a Simple Integral with Exponents

Let $f$ be differentiable in $[a,b]$. How can I show that $$\exp\left(\frac{1}{b-a} \int_a^b f(x)dx \right) \le \left(\frac{1}{b-a}\right) \int_a^b \exp(f(x)) dx $$
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2answers
135 views

How to proof the following Gronwall type inequality?

Suppose that $g,k: [0,a] \to \mathbb R$ are continuous, $a >0 $, $\,k(t) \ge 0$,$\ c(t) \in C^1([0,a])$, $\, \dot c(t) \ge 0 $ (i.e. $c(t)$ is non decreasing) and $g(t)$ satisfies $$g(t) \le ...
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0answers
23 views

An integral inequality

Suppose $K\in C[0,1]^2$, $G\in C[0,1]$ are arbitrary and given. The question is that does there exists $H\in B[0,1]$ continuous a.e. with possibly finitely many discontinuities such that $$ ...
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0answers
55 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
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1answer
34 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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2answers
68 views

An Inequality involving integration

Show that $$\int_{0}^{\pi} \left|\frac{\sin nx}{x}\right| dx \ge \frac{2}{\pi}\left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right)$$ How do I go about proving this inequality ?
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3answers
152 views

How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$

let $f(x)\in C^{1}[0,1]$ ,and such $f(0)=f(1)=0$ show that $$\left(\int_{0}^{1}f(x)dx\right)^2\le\dfrac{1}{12}\int_{0}^{1}|f'(x)|^2dx$$ I think we must use Cauchy-Schwarz inequality ...
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1answer
21 views

Show that $-2abX \le a^2Y +b^2Z\implies 4X^2 \le 4 Y Z $, $a,b \in \mathbb R $, $X;Y;Z \ge 0 $

Show that $-2abX \le a^2Y +b^2Z\implies 4X^2 \le 4YZ $ where $X,Y,Z $ are nonnegative and $a,b \in \mathbb R $. This looks almost as I could use Young's inequality, but not quiet. The above ...
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1answer
41 views

Integral inequality with sines

I am trying to show that there exists some constant $\mathcal{C}>0$ such that: $$\mathcal{C}\leq \int_0^1 |\sin (2\pi n x)-\sin (2\pi m x)|\;dx$$ For all distinct $m,n\in\mathbb{N}$. The constant ...
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28 views

Inequality involving incomplete gamma function

When trying to answer this question: Find minimum $n$ such that $1+z+\frac{z^2}{2!}+\cdots+\frac{z^n}{n!}=0$ has no answer inside the circle of radius $100$ centered at the origin I ended up in what ...
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1answer
56 views

$|f+g|^p$ Lebesgue-summable if $|f|^p$ and $|g|^p$ are

I read that the Minkowski integral inequality, which I knew for Riemann integrals on $[a,b]$, holds for Lebesgue integrals in the following form:$$\forall p\geq 1\quad\quad\Bigg(\int_X |f+g|^p ...
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1answer
30 views

How prove this integral inequality $\int_{0}^{\infty}(f(t))^2t^{-\delta}dt\le\frac{4}{(1-\delta)^2}\int_{0}^{\infty}(f'(t))^2t^{2-\delta}dt$?

Question: let $\delta\in(0,1)$, and $f\in C_{0}^{1}(R_{+})$,show that $$\int_{0}^{\infty}(f(t))^2t^{-\delta}dt\le\dfrac{4}{(1-\delta)^2}\int_{0}^{\infty}(f'(t))^2t^{2-\delta}dt$$ My idea: I ...
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30 views

Is there an inequality between the following quantities?

Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Is there an inequality between the following quantities? $$ \underset{|h|\leq ...
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57 views

Agmon and Young inequalities

I'm studying the book-Boundary control of PDEs using backstepping by M. Krstic and in Chapter 5, the authors makes a claim that I'm finding difficult to understand. We have the following two ...
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31 views

Related to Gronwall's Inequality.

The exercise is: Let $K \geq 0$, $f,g \geq 0$ continuous functions from $[a,b]$ to $\Bbb R$ and $x_0 \in ]a,b[$. Suppose that $f(x) \leq K + \left|\int_{x_0}^x f(t)g(t) \ \mathrm{d}t\right|,$ ...
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1answer
62 views

Holder Inequality

I have a problem with the demonstration of this inequality ('m following Royden) : If $p$ and $q$ are nonnegative numbers such that $\frac{1}{p}+\frac{1}{q}$ and if $f \in L^p$ and $g \in L^q$, then ...
4
votes
3answers
102 views

Prove the inequality$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$

Prove: $$\int_0^{+\infty} {\sin x \over x}dx<\int_0^\pi {\sin x \over x}dx$$ Here is my answer,but I want a different way to prove it. \begin{aligned} \int_0^{+\infty} {\sin x \over ...
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votes
2answers
28 views

Inequalities with expected value on one side and probability on the other

In a part of a proof I am following, the author states that $$\displaystyle \mathbb{E}\left[\frac{|X_n - X|}{1 + |X_n - X|}\right] \leq \epsilon + \mathbb{P}(|X_n - X| > \epsilon)$$ and ...
4
votes
4answers
95 views

Prove that $2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$

I need some starting ideas, hints for proving that $$2 \le \int_0^1 \ \frac{{(1+x)^{1+x}}}{x^x} \ dx \le 3$$ I already checked that with Mathematica that numerically says that $$\int_0^1 \ ...
10
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4answers
192 views

Prove $1.43 < \displaystyle \int_0^1 e^{x^2} \mathrm{d}x < \frac{e+1}{2}$

Prove $$1.43<\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2}$$ What I did; As I have no idea how to approach the left inequality I work with $$\int_0^1 e^{x^2} \mathrm{d}x<\frac{e+1}{2} \iff ...
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votes
1answer
34 views

How can Use Gronwall for this PDE?

I'm trying to prove this. First I tried to multiply the equation by $\phi(x,t)$ and use the Gronwall Lemma, but it didn't work. Can anyone help? Here's the problem: Given a smooth field $u:\; ...