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

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1answer
61 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
71 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 ...
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1answer
83 views

Is there an integral that proves that $\sin \tan 1\lt 1$?

I recently noted that this inequality is unbelievably sharp: $$\sin \tan 1\lt 1$$ Is there some sort of integral that can prove that this is true? This question might be of some use: Prove: $\sin ...
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1answer
15 views

A proof of $|J_{\nu}(x)|\leq x/(2\nu-1)$

I am looking for a proof of the following inequality for Bessel functions : $$|J_{\nu}(x)|\leq \frac{x}{2\nu-1} \quad \left(\text{for}~\nu>1,~0\leq x \leq \frac{\pi}{2}\right).$$ Many thanks !
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1answer
47 views

Can one control $\int(f'(x))^2$ by $\int f'(x)+f(x)$?

For a function $f(x)$ continuously differentiable and defined on [a,b] with $f(a)=f(b)=0$, can one control $\{\int_a^b[f'(x)]^2dx\}^{1/2}$ by for example ...
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2answers
55 views

Proving an Integral inequality from a given integral inequality

Problem: Let $f$ and $g$ be continuous, non-negative function on $[0, 1]$, with $$\int_{0}^{1}e^{-f(x)}dx \geq \int_{0}^{1}e^{-g(x)}dx. $$ Prove that, $$\int_{0}^{1}g(x)e^{-f(x)}dx \geq ...
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0answers
178 views

Prove $\displaystyle\int_{0}^{1} \left|\frac {f^{''}(x)}{f(x)}\right|\, dx \geq 4$ [closed]

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 $\displaystyle\int_{0}^{1} \left|\frac ...
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0answers
8 views

Relation between Karamata's and Hardy-Littlewood's inequalities

In the field of (elementary) classical inequalities one of the most famous tools is the majorization inequality due to Karamata [1] (also known as Hardy-Littlewood-Polya). In its integral version, it ...
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1answer
41 views

A sharp upper bound on discrete Young's inequality for sum with $f$ and $f^{-1}$

Problem: $f$ is a strictly monotonic and continuous function on $[0, 1]$, such that $f(0)=0$ and $f(1)=1$. Then prove that $f(\frac{1}{10})+f(\frac{2}{10})+\cdots ...
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0answers
59 views

Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$ [duplicate]

Prove that $e^x|\int_x^{x+1}\sin(e^t)dt|\le 2$. Use mean value theorem $$\int_x^{x+1}\sin(e^t)dt=\sin(e^\xi)$$ And we have $$|\sin(e^\xi)|\le\frac{2}{e^x}$$ where $\xi\in(x,x+1)$ I stuck here. ...
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1answer
17 views

For what values of $0 < p,q < \infty$ is the following inequality of integrals valid?

Let $m$ be the Lebesgue measure over $\mathbb{R}$ and let $f$ and $g$ be two nonnegative measurable functions defined on $[0,1]$ such that $f(x)g(x)\geq 1 \quad \forall x \in [0,1]$. It is not ...
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2answers
1k 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 $$ ...
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2answers
36 views

How to prove that: if $q= b+d$, then $p = a+c$?

Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$. How to prove that: if $q= b+d$, then $p = a+c$? Is there a simple way?
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0answers
21 views

Inequality verification of the ratio of two integrals involving Bessel functions

Given the following integral: $\sigma(k,\theta)=2k^2cos^2\theta\int_0^\infty J_0(2k\tau |sin\theta|) exp(-2s^2k^2\tau^{2H}cos^2\theta)) \tau d\tau$ With the following constraints ...
6
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4answers
367 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 $f\in C^{1}[0,1]$ such that $f(0)=f(1)=0$. 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
30 views

hint on exercise about weak $L^p$ space

I'm working on a problem from Grafakos, Classical Fourier Analysis. Let $(X, \mu)$ be a measure space and let $E$ be a subset of $X$ with $\mu(E) < \infty$. Assume that $f$ is in ...
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1answer
21 views

Divergence of an integral

I am sure this is a familiar example to many, as it comes up as an example of a function which belongs to $L^p$ for $p=1$ but not $L^p$ for $p < 1$, but I am having a hard time seeing why it is ...
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1answer
31 views

Integral inequality for a Cauchy exponential series product

My goal is to get an inequality $\forall t>0$ for the following integral $$ \int_0^t \left(\sum_{n=1}^\infty \exp(-n^2 t_0)\right)^2\,\mathrm{d}t_0 \le f(t). $$ The goal is to at least lose the ...
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0answers
27 views

How does this inequality imply this one?

I am having a little trouble understanding this part of a proof. There is an integral $\text{J}_{n} = \int_0^{\frac{\pi}{2}} x^2\cos^{2n}x dx$ Now, $\text{J}_0 = \frac{\pi ^3}{24} $ The part of ...
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0answers
39 views

Incomplete $\Gamma$ inequality

How to prove the following statement? For any $x > 0$, $\gamma(x+1,x) < \Gamma(x+1,x+1) < \gamma(x+1,x+1) < \Gamma(x+1,x)$ or, equivalently, $$\int_0^x t^x e^{-t} dt < ...
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1answer
23 views

Trying to find Upperbound!

Is there anyway to prove the following statement? $$\int_{0}^{T}a^T(\theta)b(\theta)d\theta \le c_1^2 \Rightarrow \int_{0}^{T}a^T(\theta)Kb(\theta)d\theta \le c_2^2$$ where $a(t),b(t)\in ...
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1answer
79 views

Find the upper bound of the derivative of an analytic function

The question is: if $f(z)$ is analytic and $|f(z)|\leq M$ for $|z|\leq r$, find an upper bound for$|f^{(n)}(z)|$ in $|z|\leq\frac{r}{2}$. My attempt: Since $f(z)$ is analytic where $|z|\leq r$, we ...
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0answers
203 views

How did he use Gronwall Lemma??

I´ve got these lines from an article: ( where $b:\mathbb{R}_+\to \mathbb{R}_+$ is non-decreasing and $(X_t)$ is an $\mathbb{R}_+$-valued process - it doesn't matter very much, I guess, anyway-.) ...
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27 views

Wirtinger's inequality

I was proving this equation in class but I ran into a problem $$\int_0^\pi u^2dx \leq \int_0^\pi (u')^2dx$$ I have $$0 \leq \int_0^\pi (u' - u \cot(x))^2 dx = \int u'-2uu'\cot(x) + u^2\cot^2(x)dx$$ ...
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1answer
21 views

Inequality for moments of sums

Suppose the random variables $X_i$ are independent and satisfy $E[X_i] = 0$. Then the following inequality holds: $$E\left[\left(\sum \limits_{i = 1}^n X_i\right)^4\right] = \sum \limits_{i = 1}^n ...
5
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1answer
456 views

continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
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1answer
32 views

How to Proceed in Solving this Equation

Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$. \begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge ...
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1answer
2k views

A function of two cumulative probability distributions with same first 2 moments

Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment (hence ...
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1answer
47 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: $$ ...
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1answer
52 views

Prove uniqueness theoremn via Gronwall inequality

A question says: Prove Theorem 1.7 (Uniqueness). Hint: suppose that $x$ and $x^*$ are distinct solutions to the same IVP (from the same initial point). Consider the function ...
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0answers
42 views

Study the variation of $\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)}$ with $q$

We define function $g(q)$ as the following: $$\frac{\Gamma(m-n^\prime+1,n^\prime q)}{\Gamma(m-n+1,n q)},$$ where $n^\prime \ge n+1$, $n^\prime \le m$. We note that $q$, $m$, $n$ $n^\prime$ are all ...
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1answer
86 views

How can show $\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2}$ [closed]

I was working on a problem and reduced it to showing the following inequality: ‎‎ $$\left(1+\frac{1}{a}\right)\left(1-\frac{1}{\sqrt{2}}\int_{0}^{1}\sqrt{1+u^a}\,du\right)<1-1/\sqrt{2};\quad ...
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0answers
21 views

Upper bound for integral on some environment of zero

I'm trying to proof an estimate that should not be too hard to proof. Let $f$ be some integrable non-negative function and $c>0$ some arbitrary constant. I claim that there exists some ...
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1answer
45 views

Where is that half coming from?

A few lessons ago, my professor proved Poincaré inequality in the following form: Let $\Omega$ be a domain contained in $\mathbb{R}^{N-1}\times(0,a)$ for some $N\in\mathbb{N},a>0$. Then for all ...
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1answer
64 views

Proving an inequality involving integrals

Let $0<a<1$, $0<b<1$, $c>0$ and $d>0$, prove the following inequality: $$\frac{1}{\frac{1}{a}+\frac{1}{b}}\geq \int_{0}^\infty\frac{1}{\frac{1}{ac\exp(-cx)}+\frac{1}{bd\exp(-dx)}}$$ ...
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1answer
497 views

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: ...
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1answer
99 views

Show $\left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx$

Given that $f: [1,e] \to \mathbb{R}$ is a continuous function, show $$ \left( \int_1^e f(x) \; dx \right)^2 \leq \int_1^e x\,f(x)^2 \; dx $$ My Attempt: At first it looked rather like a ...
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1answer
30 views

Unique weak solution to Helmholtz equation on a square

I've recently started studying the modern theory of PDEs. I studied some basic properties of Sobolev space and then started with linear elliptic PDEs. I consider the following problem: For which ...
2
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2answers
161 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 ...
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3answers
196 views

Show that $\left(\int_{0}^{1}\sqrt{f(x)^2+g(x)^2}\ dx\right)^2 \geq \left(\int_{0}^{1} f(x)\ dx\right)^2 + \left(\int_{0}^{1} g(x)\ dx\right)^2$ [closed]

Show that $$ \left( \int_{0}^{1} \sqrt{f(x)^2+g(x)^2}\ \text{d}x \right)^2 \geq \left( \int_{0}^{1} f(x)\ \text{d}x\right)^2 + \left( \int_{0}^{1} g(x)\ \text{d}x \right)^2 $$ where $f$ and $g$ ...
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1answer
27 views

Calculate limit value of sequence by inequation with integrals

I have to calculate the limit of the sequence $a_n := \sum_{k=1}^{n} \frac{1}{n+k}$ . To do so, I have to show that the following inequation is true: $\int_{n+1}^{2n+1} \frac{dx}{x} \leqslant a_n ...
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1answer
36 views

How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$?

$\Omega\subset \mathbb R^n$ is bounded and open. $u,v\in H_0^1(\Omega)$. $Du$ is gradient of $u$. How to show that $\int_\Omega \sum\limits_{i,j=1}^n u_{x_i}v_{x_j} dx\le C\int_\Omega |Du||Dv| dx$ ?
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2answers
80 views

Prove a function is in $L^2[0,1]$

If $f\in L^2[0,1]$, and $$g(x)=\int_0^1\frac{f(t)\mathrm dt}{|x-t|^{1/2}},\quad x\in[0,1],$$ show that $\|g\|_2\le2\sqrt2\|f\|_2$. I tried Minkowski's integral inequality (with $p=1/2$, so ...
0
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1answer
29 views

An integral inequality with sequence

My works lead to the true of the following inequality: For any $p>0$, there exist a constant $C_p>0$ which depends only on $p$, such that for any nonnegative sequence $(x_k)_{k\ge1}$ and for ...
1
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1answer
44 views

Weak Law of large numbers involving a sequence and random variable

During one of our Information theory classes, the professor constructed the following set: $$T_\delta = \left\{\mathbf{y} \in \mathbb{R}^n: \frac{\sum_{i=1}^ny_i^2}{n} \leq P + ...
3
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2answers
54 views

Show that $\int_{|x|<R}\frac{|f(x)|}{|x|}\mathrm dx=o(R),\quad R\to\infty.$

Suppose $f\in L^3(\mathbb R^3)$. Show that $$\int_{|x|<R}\frac{|f(x)|}{|x|}\mathrm dx=o(R),\quad R\to\infty.$$ First, I try to show that for a fixed $R_0$, ...
0
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1answer
22 views

Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$

$a=a(x), b=b(x)$ are elements of $L^p(\Omega)$, $\Omega$ is bounded open subset of $R^n$. Whether $||a||_{L^p} +||b||_{L^p}\le||\sqrt{a^2+b^2}||_{L^p}$ ?
1
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1answer
62 views

Evaluate $\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$ or find good upper bound.

Is it possible to evaluate or at least to estimate the following integrals? $$\int_0^n \frac{|\cos nx| }{2+x^2+\sqrt{|\cos nx|}} dx$$ and $$\int_0^n \frac{|\cos nx| }{2+x+\sqrt{|\cos nx|}} dx$$ I ...
1
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1answer
34 views

Show something converge to infinity strongly

Show that if $ Var(Y_n)=1 $ and $ \mu_n\to \infty $ quickly enough so that $ \sum_{n=1}^{\infty}\frac{1}{\mu_n^2} $ is finite, then $ p[Y_n\to \infty]=1. $ I know that I need to use Chebyshev's ...
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1answer
45 views

A function given as an integral is uniformly continuous provided the integrand is uniformly continuous

I need to show that this inequality holds: $| \int_{0}^{1} (h \nabla f(x+sh-y) -h \nabla f(x-y)) ds | \leq |h| \varepsilon(|h|)$ For a function $\varepsilon$ which verifies $\varepsilon (|h|) ...