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

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Prove or disprove this inequality

Let $p, q, a$, and $b$ be natural numbers such that $p<q$, $1<b<a$ and $b\nmid a$. Is is true that $(bp+aq)^3> (a^3+b^3)q^3$? This is what I tried: expanding the left-hand side, we ...
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1answer
15 views

Determining Bounds to calculate mass

Let $E$ be the solid region defined by the inequalities $x \ge 0$, $0\le z \le \sqrt(x^2 + y^2)$, $x^2 + y^2 + z^2 \le 4$ Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the ...
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43 views

Inequality: product of integrals

Context: Proving integral inequalities about posterior distributions following different sequences of binary signals. The proofs come down to the following inequalities. Let $\psi(x)$ be a concave ...
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71 views

Dominance between two functions

Let two functions $f(z)$ and $g(z)$ with $z\in[0,c]$ with $c$ a constant such that $c<b$. I'd like to check whether $f(z)-g(z)>0$. I've tried to set $f(z)$ to its minimal value and $g(z)$ to its ...
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40 views

Rewrite order of $\int_0^1\int_0^{\sqrt{y}} \int_y^1 \, dz \, dx \, dy$ to $dx\,dy\,dz$ and $dy\,dz\,dx$

I need to change the order of $$\int_0^1\int_0^{\sqrt{y}}\int_y^1\,dz\,dx\,dy$$ to $dx\,dy\,dz$ and $dy\,dz\,dx.$ I can extract the inequalities to get $1≤z≤y$, $0≤x≤\sqrt y$, $0≤y≤1$, but I get ...
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1answer
21 views

Inequality releating squared absolute value of an integral to the integral of the squared absolute values of the integrand

Is this inequality $\left| \int_{0}^{x} f(t) \ dt \right|^2 \leq \int_{0}^{x} |f(t)|^2 \ dx$ true for $x\in [0,1]$. In case it is how to prove it? If there is no square in both sides it is easy since ...
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55 views

Integral inequality with gradient

Let $\psi \in C_0^{\infty}(\mathbb{R}^3)$. How to prove (or where I can find this proof) that $$\int_{\mathbb{R}^3}\frac{1}{4r^2}|\psi(x)|^2d^3x\le \int_{\mathbb{R}^3}|\nabla\psi(x)|^2d^3x$$ ? ...
4
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178 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
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51 views

Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem

This question spanned from a previous interesting one. Let $k$ be a real number greater than $2$ and $$\varphi_k(\xi) = \int_{0}^{+\infty}\cos(\xi x) e^{-x^k}\,dx $$ the Fourier cosine transform of a ...
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1answer
26 views

Absolutely continuous function inequality

For 1 $\le$p<$\infty$, show that if the absolutely continuous function $F$ on $[a,b]$ is an indefinite integral of an $L^p[a,b]$ function, then there exists an $M>0$ such that for any partition $...
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182 views

Bounds on $f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \, dx}{ \int_0^\infty \cos(b x) e^{-x^k}\, dx}$

Suppose we define a function \begin{align} f(k ;a,b) =\frac{ \int_0^\infty \cos(a x) e^{-x^k} \,dx}{ \int_0^\infty \cos(b x) e^{-x^k} \,dx} \end{align} can we show that \begin{align} |f(k ;a,b)| \...
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45 views

$\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$

We consider two i.i.d. random variables $U_1$ and $U_2$ such that $\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$ and $\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any $c > 0$ the ...
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1answer
26 views

Integral Inequality with Monotonic Function

Problem: For continuous, either both increasing or both decreasing functions $f, g$ on $[a, b]$, suppose that $p(x)$ is continuous and positive. Prove that $$\int_a^bp(x)f(x)dx \int_a^bp(x)g(x)...
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63 views

Proving a Definite Integral Inequality without Geometrical Intuition

I solved an integral inequality problem using geometrical methods. However, I just cannot satisfy with them and want a without-geometrical-intuition proof, and I couldn't find one. Proof the ...
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17 views

Integral and differential inequality

I have integral and differential inequality $y'(t)<Ch^{k+1}+\int_0^ty(s)ds+y(t)$ where $C,h$ are constants and $y$ is positive function with y(0)=0 My goal is to prove $y(t_F)<Ch^{k+1}$ ...
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1answer
39 views

A Young's inequality used to bound curvature terms

I've been having a look at the Gage and Hamilton's The Heat Equation Shrinking Convex Plane Curves (here). In particular I've been working in the Lemma 4.4.2 and some further results where they find ...
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252 views

integral inequality for $f(x)$ and $f(\sqrt{x})$

Show that if $f(x)\in [0;1]$, $f\in C$ and $\int\limits_{1}^{+\infty}f(t)dt=A$ then $\int\limits_{1}^{+\infty}tf(t)dt>\frac{A^2}{2}$ I only have noticed two small things: If $A=1$ inequality is ...
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30 views

Kind of Gronwall Inequality

Does somebody knows if it is possible to obtain an inequality (like for Gronwall inequality) on $f$ if $f$ verify $$ f(t) \leq A+\int_0^{2t} g(s)f(s) ds $$. Where $f$ and $g$ are as smooth as ...
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1answer
33 views

For what value of $x$: $ n^ {(x+1)} + n^ {2x} < n^2$ ? Where, $0\leq x <1$ and $n$ is constant integer value & $n>1$.

How to find the optimal value of $x$ and what is the relation between $x$ and $n$ i.e. How to get dependency between $x$ and $n$? As per my understanding, solution should be in term of $n$ like like ...
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44 views

Inequality of Scalar Product involving derivative

I am stuck trying to reach an (in)equality... Let $\Omega \in \mathbb{R}$ and $f=f(t,x): \mathbb{R} \supset[0,T] \times \Omega \rightarrow \mathbb{R}$ be an element of $$\mathcal{W}:=L_{_t}^2(0,T,H_{...
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52 views

A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
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92 views

Integral inequality :$\int_0^1(f'(x))^2dx\geq 32\int_0^1(f(x))^2dx + 16\left(\int_0^{\frac{1}{2}}f(x)dx-\int_{\frac{1}{2}}^1f(x)dx\right)^2$

Assume $f:[0,1]\to \mathbb{R}$ is differentiable and $f'$ is integrable. Given $f\left(\frac{1}{4}\right)=f\left(\frac{3}{4}\right)=f(1)-f(0)=0$, then prove that $$\int_0^1(f'(x))^2dx\geq 32\int_0^1(f(...
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61 views

Integral inequality involving $f(x),\, x\,f(x),\, f(x)^2$

Let $f\colon[-1,1]\to\mathbb{R}$ be a continuous function. Prove that $$ 2\int_{-1}^{1}f(x)^2\: dx - \left(\int_{-1}^{1}f(x)\: dx\right)^2 \ge 3\,\left(\,\int\limits_{-1}^{1}x\,f(x)\: dx\right)^2 $$ ...
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Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where $C$...
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124 views

Is the Riemann integral of a strictly smaller function strictly smaller?

We all know that if $f\leq{}g$ in $[a,b]$ then $$ \int_a^bf\,dx\leq\int_a^bg\,dx $$ now, imagine that we have $f<g$, is it true that $$ \int_a^bf\,dx<\int_a^bg\,dx $$
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1answer
28 views

Prove, for every $l \geq 3$ , the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds

I need to prove that for every $l \geq 3$, the $\Big( 1- \dfrac{1}{2 \cdot l}\Big)^{2 \cdot l} < \dfrac{1}{e}$ holds. ($l$ is integer) This is what I tried so far. $$ \begin{align} x &= \...
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1answer
85 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
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1answer
59 views

Explain inequality of integrals by taylor expansion

I try to understand why the following inequality holds. $$\left|\int_{|y|<1} e^{iuy}−1−iuy\ \, dy \right| \le \frac{1}{2} \cdot \int_{|y|<1} |uy|^2\ \, dy$$ Due to a hint I'm pretty sure, ...
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1answer
51 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} E[(...
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28 views

Poincare type inequality on unit square: can you improve on my constants?

I am trying to bound $\int_{[0,1]^2} u^2$ in terms of its gradient and boundary integrals, $\int_{[0,1]^2} |\nabla u|^2$, $\int_{\partial[0,1]^2} u^2$, with the best possible constants. So far I ...
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89 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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65 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 \int_{0}^{1}...
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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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42 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 f(\frac{9}{10})+f^{-1}(\frac{1}{10})+...
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50 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 $\int_a^b|f'(x)|dx+\{\int_a^b|f(x)|^pdx\}^{1/p}...
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61 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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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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23 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 $0.5<H<1$...
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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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1answer
38 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 $L^{p,\infty}(...
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1answer
23 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
33 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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29 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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42 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 < \...
2
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1answer
24 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 \mathbb{R}^...
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1answer
106 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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29 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
22 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 E[...
4
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210 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-.) $...