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

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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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57 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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16 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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38 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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248 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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24 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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31 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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35 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 $$L^2(0,T,H^1(\Omega)):= \left\{...
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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 $...
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Integral inequality :$\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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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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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
84 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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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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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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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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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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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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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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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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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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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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22 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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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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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 < \...
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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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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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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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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[...
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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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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 (z-...
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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 $\nu(t)=||x(t)-x^*(t)||...
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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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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 a>0....
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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
47 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
100 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 Cauchy-...
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1answer
30 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
66 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
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 ...
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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 $R_0$, $\int_{|x|<R_0}\frac{|f(x)|}{|x|}\...