Fie
\( \mathbb{H} \) un spatiu Hilbert real si
\( u:[0,1]\to\mathbb{H} \) derivabila cu derivata continua astfel incat
\( u(0)=u(1)=0 \). Aratati ca
\( \int_0^1 |u(t)|^{2}dt\leq\frac{1}{\pi^2}\int_0^1 |u\prime(t)|^{2}dt -\left(\int_0^1|u(t)|\cot\pi tdt\right)^{2} \).
P.S. Chair sunt curios daca acesta rafinare exista pe undeva.
Eu am cautat in articolele lui Osermann si Ravi Agarwall, dar nu am gasit decat variante ale inegalitatii lui Wirtinger. Un alt articol interesant ar fi si acesta de
aici.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
