On a generalized Wirtinger inequality

Let $\alpha( p,q,r) =$inf{$\frac{|| u'||_p}{||u||_q}:u\in W_{p e r}^{1,p}( -1,1) $\{$ 0$}, $\int_{-1}^1|u|^{r-2} u=0$} . We show that $\alpha( p,q,r )=\alpha ( p,q,q)$ if $q\leq rp+r-1$ $\alpha( p,q,r) ( 2r-1) p$ generalizing results of Dacorogna-Gangbo-Subia and others.