Sur une généralisation de l'inégalité de Wirtinger. (A generalization of Wirtinger's inequality) (Q1185268)

Let \(\alpha_ 1=\alpha_ 1(p,q)=\min\left\{{\| u'\|_{L^ p}\over\| u\|_{L^ q}}| u\in W^{1,p}(- 1,1)\backslash\{0\},\;u(-1)=u(1),\;\int^ 1_{-1}u| u|^{q- 2}=0\right\}\), \(\alpha_ 2=\alpha_ 2(p,q)=\min\left\{{\| u'\|_{L^ p}\over\| u\|_{L^ q}}| u\in W^{1,p}(- 1,1)\backslash\{0\},\;u(-1)=u(1),\;\int^ 1_{-1}u=0\right\}\). We compute explicitly \(\alpha_ 1\) and we show that for \(q\leq 2p\), \(\alpha_ 1=\alpha_ 2\), while for \(q\) sufficiently large \(\alpha_ 2<\alpha_ 1\).