On a generalized Wirtinger inequality

Abstract

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