In Rudin's Real and Complex Analysis, in Theorem 11.16 page 239-240 he says that if $f\in L^p(T)$ (T is the unit circle) and if $u=P[f]$, where $P[f]$ is the Poisson integral of $f$, then $||u_r||_p\le||f||_p ,0\le r\le1 (u_r(e^{\theta i})=u(re^{\theta i}))$.
In the proof he says that
$|u_r(e^{i\theta})|^p\le1/2\pi\int_{-\pi}^{\pi}|f(t)|^pP_r(\theta-t)dt $
and that this inequality can be deduced from Holder. Can someone explain how does this follow from Holder?
Thanks.