query: How to prove $\int_{-\pi}^\pi (f(x))^2dx\le \int_{-\pi}^\pi (f'(x))^2dx$
positive_passages.docid: 156599
positive_passages.text: Wirtinger Inequality: $\int_0^1 f'^2 \geq \pi^2 \int_0^1 f^2 $
negative_passages.docid: 41191
negative_passages.text: How to prove this $\pi$ formula?
negative_passages.docid: 76408
negative_passages.text: Show that $\int_0^\pi f(\sin x)\,\mathrm{d}x = 2\int_0^{\pi/2}f(\sin x) \, \mathrm{d}x$
negative_passages.docid: 116056
negative_passages.text: Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
negative_passages.docid: 8358
negative_passages.text: Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $
negative_passages.docid: 118884
negative_passages.text: The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$
negative_passages.docid: 97253
negative_passages.text: How prove $e^x|f(x)|\le 2$ if $f(x)=\int_{x}^{x+1}\sin{(e^t)}dt$
negative_passages.docid: 157426
negative_passages.text: Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 10675
negative_passages.text: Proving that $\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^2dt=\sum_{n=0}^{\infty}|a_n|^2r^{2n}$
negative_passages.docid: 50038
negative_passages.text: Proving that $e^\pi=e^{-\pi}$
negative_passages.docid: 30385
negative_passages.text: Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
negative_passages.docid: 37015
negative_passages.text: Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
negative_passages.docid: 51837
negative_passages.text: Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
negative_passages.docid: 154264
negative_passages.text: $f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$
negative_passages.docid: 61162
negative_passages.text: Show that $\left(\int_0^1 f(x)\,dx\right)^2\leq2\int_0^1x\, f(x)^2\,dx $
negative_passages.docid: 114982
negative_passages.text: How do I get that $|\int_a^b f| \le \int_a^b |f|$?