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

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7
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134 views

How to prove Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have $$(f(t))^4\le ...
5
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114 views

Functional inequality with a strong RHS

Consider a continuous function $f:[0,1]\to\mathbb{R}^{+}$. Show that $$\int_0^1 f(x)dx-\exp\left(\int_0^1 \log(f(x)) dx\right)\le \max_{0\le x,y\le 1}\left(\sqrt{f(x)}-\sqrt{f(y)}\right)^2$$ I ...
4
votes
0answers
108 views

A mixture with ingredients of two equivalences with Riemann Hypothesis

Let $f(x)=x\cdot(\log x)^x$ for $x\geq 2$, then integrating $\log f(x)=\int_2^x f'(t)/f(t)dt$, it is easy to prove the first statement of following, and directly if we put $x=e^H_n$ and add $H_n$ the ...
4
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177 views

hard integral inequality with $e^{x^2}$

a) prove the convergency of $$ \int_0^{\infty} \dfrac {\cosh x \cdot \sin x} {x\cdot e^{x^2}}$$ b) prove the inequality $$ \int_0^{\frac {7\pi} {12}} \dfrac {\cosh x \cdot \sin x} {x\cdot ...
3
votes
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41 views

Inequality involving homogenous function of degree -1

Let $p\in[1,\infty]$. For $f\in L^p(0,\infty)$ we define $Tf:x\mapsto \int_0^\infty K(x,y)f(y)\,dy$ where $K$ is homogenous of degree $-1$, i.e. $K(\lambda x,\lambda y) = \lambda^{-1} K(x,y)$ for ...
3
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0answers
115 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 ...
3
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105 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 ...
2
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188 views

Integral inequality - lower bound on $L^1$ norm.

I was wondering if one can make an estimate of form: Assume $f\in C^\infty(\overline{\Omega})$ where $\Omega$ is a bounded domain in $\mathbb{}R^d$. Is there a constant $C>0$ independent of $f$ ...
2
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0answers
41 views

Question about the assumption of a version of Grönwall's inequality.

According to Wikipedia, A version of Grönwall's inequality for the integral of continuous functions is the following: Let $I$ denote an interval of the real line of the form $[a,\infty)$ or $[a,b]$ ...
2
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54 views

integral inequality involving $\sup|f'|$

Let $f:[0,1]\rightarrow \mathbb R$ be continuous function differentiable on $(0,1)$ with property that there exists $a \in (0,1]$ such that $$\int_{0}^a f(x)dx=0$$ Prove that $$\left|\int_{0}^1 ...
2
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46 views

Proving inequality that bounds the sum of norms with the norms of sums (plus additional terms)

I am struggling with showing the following for finite $\delta>0$ and any $g\in\mathcal{G}_1\times...\times\mathcal{G}_k$: ...
2
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0answers
97 views

Inequality involving double integral

There's a function $g(x,y):\mathbb{R^+\times \mathbb{R}^+\rightarrow \mathbb{R}^+}$ with $g_1(x,y)>0$, $g_2(x,y)>0$, and $g_{12}(x,y)>0$. I conjecture that $$\int ...
2
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42 views

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 ...
2
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0answers
122 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 ...
2
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81 views

Gagliardo-Nirenberg-Sobolev inequality

There is one step of the textbook's proof that I wish could be clarified. Pages 277-278 of PDE Evans, 2nd edition, says: Integrate this inequality with respect to $x_1$: \begin{align} ...
2
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63 views

Integral inequality related to derivation

While trying to understand a proof, i have stumbled upon the following statement: Let $f \in L^p(a,b)$ be a $p$-integrable function. Then the inequality $$\liminf_{s \rightarrow t} \frac{1}{t-s} ...
2
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229 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
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41 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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23 views

Singular Gronwall type inequality

I am looking for a proof or refrence for the following Gronwall-type inequality: Let $ \varphi (t,s) $ is a continous function for $0 \leq s < t \leq T$. If the following inequality holds: $$ ...
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21 views

Inequalities with the Integral Test

a) Use the proof of the integral test to show that $\ln(n!)\ge n\ln(n)-n+1$ for $n>1$ b) Use part (a) to show that $\ln(n!)\ge n\ln(n)$ for $n\ge 10$ I was able to solve part a) but not ...
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13 views

An inequality between functions in Munkenhoupt class

I'm currently studying the Munkenhoupt class (also called $A_p$ class), and I'm stumbled upon the proof of the following property: Given $1 < p <\infty$, and $w\in A_p(R^n)$ (note that we have ...
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15 views

Is $\int_1^2 \int_{(3/2-1/x_1)^{-1}}^2 \cdots \int_{[(n+1)/2-\sum_{k=1}^{n-1} 1/x_k]^{-1}}^2 \prod_{k=1}^n f(x_{k-1},x_k) dx_1 \dots dx_n \geq cq^n$?

Let $f\colon (1,2)\times (1,2) \to \mathbb{R}$ be a Lebesgue measurable, bounded and non-negative function such that $$ \int_1^2 f(x,y) dy = 1, \qquad x \in (1,2). $$ Moreover, assume that for any ...
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37 views

Question about the interpret of Picone inequality for non-regular functions.

Assume $\Omega \subset \mathbb{R}^N$, $ N>4 $ is open. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
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41 views

Hölder type inequality or similar or general inequality for integral of product

It is well known that for two given functions $f,g:\mathbb{R}^d \rightarrow \mathbb{R}^d$ such that $fg \in L^1(\mathbb{R}^d)$ and $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ with ...
1
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0answers
39 views

“Transference” argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
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25 views

Step function scalar product inequality

I would like to prove the following inequality: $$\langle f,\frac{|N.+1|}{N} \rangle^2 \leq \langle f,. \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i ...
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30 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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38 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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74 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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48 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
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47 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
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50 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
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73 views

Extending by zero a Sobolev function

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set and $u\in W_0^{1,2}(\Omega)$. Define $B_R=B(x_0,R)$ for $x_0\in\partial\Omega$ and consider $\tilde{u}=u\chi_{\Omega\cap B_{2R}}$. Do we need some ...
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52 views

Does Jensen's inequality become stricter with respect to the right boundary point?

Let $f(x)>0$ for all $t\in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ be a density. I will truncate the density to the finite interval $[a,b]$ and will eventually be taking ...
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0answers
56 views

inequality question with integrals

There is a question that, I think, has a definite answer, but I can't figure it out. Given are three real valued functions, $f$,$g$, $w$, of a real variable $x$. The functions are non-negative, i.e., ...
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315 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
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156 views

When $\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$

Under what conditions on $f(x)$ the following equation holds? $$\int_a^b\left(f(x)\right)^kdx=\left[\int_a^bf(x)dx\right]^k$$ with $k\in\mathbb{N}$ and $k\gt1$. I know the following inequality holds: ...
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42 views

Estimation of a scalar product

I encountered the following, which shouldn't be that hard, but I can't get my head around it. The problem is the following estimate (part of a bigger equation, but here's just the difficult part): ...
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132 views

Inequalities of integrals of periodic functions

I have a function that has a shape similar to $\sin(x)^2$ (could be periodic extensions of $(x/(\pi/2))^2$ defined between $-\pi/2$ to $\pi/2$ for example). Let's call it $g(x)$. I want to show that ...
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18 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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0answers
10 views

An Integral Inequality over the space of Probability Distribution with a Parameter

Suppose $P$ is the set of functions where $p\in P: R^{+2}\to R^+$ and $p(t,s)$ is differentiable in $t$. $\forall t, p(t,\cdot)$ is a probability distribution on the positive axis $s\in [0,\infty)$, ...
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60 views

Integral inequality of transformed integrand with second order stochastic dominance flavor

Let $f,g : [0,1] \rightarrow [0,1]$ be two functions such that for all $x \in [0,1]$ $\int_0^x f(t) dt \geq \int_0^x g(t) dt$ and $\int_0^1 f(t) dt = \int_0^1 g(t) dt.$ Can I conclude that ...
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0answers
17 views

Stochastic Dominance and Jensen's Inequality

Let $f(x;z)$ be a pdf for the random variable $x$ and define $$g(z)=\frac{\int x^2 f(x;z)dx}{\left (\int x f(x;z) \right )^2}$$ Assume that $z'>z$ implies that $f(x;z')$ first-order stochastically ...
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22 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
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37 views

I attempt to combine Abel's summation with Hardy's inequality

Let $a(n)$ be a sequence of real numbers, $A(x)=\sum_{n\le x}a(n)$, with $A(x)=0$ if $x<1$ and $G(x)=\int_0^x g(t)dt$, with $g(t)\ge 0$ integrable on $[0, \infty)$, $p>1$ (is a requeriment for ...
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0answers
18 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
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0answers
21 views

how to separate expectation of the product of two r.v.s

Here are two non-negative random variables $X,Y$. $X$ has finite moment of order 2, and it could have infinite moments of higher order. $Y$ has finite moment of any order. They are correlated, but ...
0
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0answers
41 views

Step function integral inequality

I would like to prove the following inequality: $$\langle f,Id \rangle^2 \leq \langle f,1 \rangle^2$$ where $f$ is a step function of the form $f(s)=\sum\limits_{i=1}^N f_i \mathbb{1}_{I(i)}(s)$, ...
0
votes
0answers
15 views

Bounding oscillating integrals over short intervals

Let $f: \mathbb{R} \to \mathbb{C}$ be written as a product $$f(x)=\vert f(x) \vert \mathrm{exp}(i \,\mathrm{arg} f(x)),$$ and suppose that $\mathrm{arg} f$ is chosen so as to be continuous. Suppose ...
0
votes
0answers
32 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 ...