A sharp weighted Wirtinger inequality

Ricciardi, Tonia

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 1, pp. 259-267

Résumé

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality $$ \int_{0}^{2\pi}\gamma^{p}\omega^{2}\leq C \int_{0}^{2\pi}\gamma^{q}\omega'^{2} $$ where $\gamma$ is bounded above and below away from zero, $w$ is $2\pi$-periodic and such that $\int_{0}^{2\pi}\gamma^{p}\omega=0$, and $p+q\geq 0$. Our result generalizes an inequality of Piccinini and Spagnolo.