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We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality $${\int }_0^{2\pi }{\gamma }^p{\omega }^2\le C{\int }_0^{2\pi }{\gamma }^q\omega ^{\prime }{}^2$$ 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 $p+q\ge 0$. Our result generalizes an inequality of Piccinini and Spagnolo.