# Tagged Questions

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### How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$ [duplicate]

Let $f$ be $C^1$ in $[-\pi, \pi]$ and satisfies $\int_{-\pi}^\pi f(x)dx=0$, periodic boundary condition. Then, prove that $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$. I try to prove ...
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### Show $\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$

Assume $f$ and $g$ are monotonically increasing on $[0,a]$, Show that $$\displaystyle\int_0^af(x)g(x)dx\ge\int_0^af(a-x)g(x)dx$$ If I differentiate both sides w.r. to $a$ then; ...
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### 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 ...
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### Need help filling in the details of proof of Jensen's Inequality

In a book on PDEs that I'm reading, I am trying to fill in the details of the proof of Jensen's Theorem, and am having a little trouble with the algebra. Here is the statement of Jensen's Theorem in ...
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### Find $\inf_{f > 0} T_f := \left(\int_A f \, d\mu\right)\left(\int_A \frac{1}{f} \, d\mu\right)$

This exercise gives me trouble: Let $F$ denote the collection of measurable functions which are positive $\mu$-a.e. and let $A \in \mathbb X$ satisfy $0 < \mu(A) < \infty$. For $f \in F$ let ...
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### Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
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### Integral inequality $\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$

How to prove this inequality $$\int_0^1\log \left(f(x)\right)dx\leq \log\left(\int_0^1f(x)dx\right)$$ for $f>0$.
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### Wirtinger's inequality in higher dimension

Wirtinger's inequality for one-dimensional functions states that if $f(x)$, $f'(x) = \frac{df(x)}{dx}$ $\in$ $\mathcal{L}^2(a,b)$ and either $f(a) = 0$ or $f(b) = 0$ then \int_{a}^{b} ...
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### How to prove a duality of $L^p$ spaces? [duplicate]
Let $(\Omega,\Sigma,\mu)$ be a finite measure space and $f:\Omega\longrightarrow \mathbb{R}$ be a measuable function. Let $1\leq p< \infty$ and $1/p+1/q=1$. Prove that the following are equivalent: ...