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

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

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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3answers
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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1answer
21 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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1answer
53 views

one inequality involving two stochastic processes [on hold]

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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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 $y(t_F)<Ch^{k+1}$ ...
9
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2answers
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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1answer
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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25 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
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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1answer
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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0answers
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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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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2answers
90 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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2answers
57 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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23 views

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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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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3answers
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
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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0answers
19 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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1answer
62 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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72 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
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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1answer
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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64 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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183 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 {f^{''...
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10 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
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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60 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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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 $$ \int_a^b\Big|\...
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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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22 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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385 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 $$\int_{0}^{1}|...
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1answer
34 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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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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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 < \...
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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
97 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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209 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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28 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[...
5
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1answer
475 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 (z-...
7
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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
66 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 $\nu(t)=||x(t)-x^*(t)||...