A Poincaré type inequality with three constraints

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $\inf_{u\in\mathcal{W}}\frac{\displaystyle\int_{-π}^π[(u')^2-u^2]dθ}{\displaystyle\left[\int_{-π}^π|u| dθ\right]^2}$where $u\in \mathcal{W}$ is a $H^1(-π,π)$ periodic function, with zero average on $(-π,π)$ and orthogonal to sine and cosine.

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