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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1answer
451 views

Are there any interpretations for the Gronwall's inequality in view of comparison theorem?

One form of the Gronwall's inequality is that If $\alpha(x),u(x)$ are non-negative continuous functions on $[0,1]$, and $$\forall x\in [0,1], u(x)\leq ...
2
votes
1answer
44 views

Help understanding an application of Jensen's inequality

This is from the book Pattern Recognition and Machine Learning by Christopher Bishop. The author states the following form of Jensen's inequality: $f\left(\int{xp(x)dx}\right) \leq \int{f(x)p(x)dx}$ ...
1
vote
1answer
49 views

Struggling to prove inequality

I've been given to following inequality to prove: (The hint given was not to evaluate the integral) \begin{equation*} \frac{1}{4} \leq \int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{sin(x)}{x}dx\leq ...
1
vote
1answer
39 views

Upper bound for integral over boundary in terms of integral over interior

I've encountered quite some papers in which it is simply assumed that $\exists C>0 : \left(\displaystyle{\int\limits_{\Gamma}}((\nabla v)\cdot \hat{\bf{n}})^2d\Gamma\right)^{\dfrac{1}{2}}\leq ...
1
vote
1answer
76 views

Hausdorff-Young inequality

Let $1<p\leq2\leq q \leq \infty$ and let: $$ \frac{1}{p} + \frac{1}{q}=1 $$ prove that for all finite Abel groups and all functions $f:\mathbb{A}\rightarrow \mathbb{C}$ Hausdorff-Young ...
1
vote
1answer
165 views

If the weighted Lp norm of a measurable function is finite, is the weighted Lp norm of the antiderivative also finite?

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function such that $$ \int_{-\infty}^{\infty} |f|^p e^{-x^2} dx < \infty. $$ Define $g : \mathbb{R} \rightarrow \mathbb{R}$ to be $$ ...
0
votes
1answer
12 views

Absolute value inequality for Pettis integral

Let $f:[a,b]\rightarrow E$ be absolutely continuous and Pettis integrable, i.e. there exists $I_f\in E$ such that $x^*(I_f)=\int x^*\circ f$ for $x^*\in E^*$. Because $f$ is absolutely continuous, ...
0
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1answer
27 views

A relationship between continuous function and its reciprocal

Let $f$ is a continuous, positive function in $[0,1]$ and let $M$ and $m$ are the maximum and minimum values of $f$ in $[0,1]$. It is easy to see that $$ ...
0
votes
1answer
29 views

Existance of solution of $Ax < b$

How to check if the inequality $Ax < b$ admits at least one solution. Entries of $A$, $x$ and $b$ are taken in $\mathbb{Z}$
0
votes
1answer
36 views

Integral Inequality involving the Euclidian Norm

I have spent several hours trying to establish the inequality shown in the attached photo. Here we assume that $\vec{r}(t)$ is a vector function in $R^n$, and is integrable on $[a,b]$. I am in need of ...
5
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0answers
107 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
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0answers
79 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 ...
4
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0answers
226 views

Hölder regularity of the simple layer heat potential (question on the proof)

Let $G(t,x)$ be the fundamental solution of the heat equation, with $t\in\mathbb{R},x\in\mathbb{R}^n$. In the book "Linear and Quasilinear Equations of Parabolic Type" by O.Ladyzhenskaya, ...
3
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0answers
114 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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0answers
70 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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0answers
31 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
38 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
votes
0answers
28 views

log-sobolev inequalities for infinite measures.

I was wondering if we could have log-sobolev inequalities for infinite measures, most notably Lebesgue measure. I presume this is false, but I haven't been able to construct one. I tried playing ...
2
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0answers
60 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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0answers
61 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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0answers
56 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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0answers
198 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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0answers
26 views

Bound on the integral of a function with multiple zeros

This is a follow-up to this If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that ...
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0answers
11 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 ...
1
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0answers
24 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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0answers
36 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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0answers
48 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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0answers
41 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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0answers
32 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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0answers
46 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 ...
1
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0answers
57 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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0answers
48 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 ...
1
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0answers
55 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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0answers
214 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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0answers
154 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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0answers
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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0answers
123 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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0answers
15 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 ...
0
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0answers
14 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
22 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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0answers
25 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
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0answers
13 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
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0answers
14 views

Upper-bounds to $B_z(a,b)$

Is there any standard technique to produce nice upper-bounds to the incomplete beta function $$B_z(a,b)=\int_0^z t^{a-1} (1-t)^{b-1} dt \,?$$ Disclaimer: this question is intentionally not too ...
0
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0answers
27 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 ...
0
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0answers
24 views

Sufficient conditions for inequalities

Assume $f$ and $g$ are two Lebesgue integral functions on interval $[a,b]$. Do we have any nontrivial sufficient conditions on $f$ and $g$ such that $$\forall a\leq x<y\leq b,\quad ...
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0answers
32 views

Is there an inequality between the following quantities?

Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Let $f\in L_{p}([0,1])$ and $[a,c]\subset [0,1].$ Is there an inequality between the following quantities? $$ \underset{|h|\leq ...
0
votes
0answers
93 views

Agmon and Young inequalities

I'm studying the book-Boundary control of PDEs using backstepping by M. Krstic and in Chapter 5, the authors makes a claim that I'm finding difficult to understand. We have the following two ...
0
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0answers
28 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
0
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0answers
43 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: $$ ...
0
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0answers
79 views

Median-median inequality

An elementary result from Chebyshev's theorem is that the median and mean of a random variable do not differ by more than one standard deviation. I'm curious if there is a similar result for ...