A sharp weighted Wirtinger inequality

Abstract

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2π}γ^pw^2\le C\int_0^{2π}γ^qw'^2 \] where $γ$ is bounded above and below away from zero, $w$ is $2π$-periodic and such that $\int_0^{2π}γ^pw=0$, and $p+q\ge0$. Our result generalizes an inequality of Piccinini and Spagnolo.