1
vote
0answers
27 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 ...
2
votes
1answer
68 views

Cauchy-Schwarz-like inequality of integrals

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$ I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$ so if we ...
3
votes
2answers
57 views

Proof about boundedness of $\rm Si$

$\def\Si{{\rm Si}}$ I want to prove the boundedness of $$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$ as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to ...
0
votes
2answers
302 views

How to prove equality from poincare inequality?

Let $$D = \{y \in C^1(0,1) : y(0) = y(1) = 0\}$$ Suppose there exists a $C_0$ such that $$\int_{0}^{1} y^2 \ dx \leq C_0 \int_{0}^{1} (y')^2 \ dx$$ for all $y \in D$, and for all $C < C_0$ ...
1
vote
2answers
82 views

Prove with integration the inequality $e(\frac{n}{e})^n < n! < n*e(\frac{n}{e})^n$

Prove with integration the inequality, I need some advice about how to start prove it. I know that if function is Monotonically increasing function so : $$ f(1)+\int^n_1f(x)dx\leq ...
2
votes
1answer
538 views

question 9 - chap 5 evans PDE

The question is : Integrate by parts to prove : $$\int_{U} |Du|^p \ dx \leq C \left(\int_{U} |u|^p \ dx\right)^{1/2} \left(\int_{U} |D^2 u|^p \ dx\right)^{1/2}$$ for $ 2 \leq p < \infty$ ...
0
votes
1answer
136 views

Identifying inequalities for 3D region.

I am having some difficulty identifying the bounds of a three-dimensional region. I am asked to evaluate $\iiint_R (xz+3z)dV$, where $R$ is the region bounded by the cylinder $x^2 + z^2 = 9$ and the ...
5
votes
1answer
766 views

Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...