A Poincaré type inequality with three constraints

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