The rate of increase of mean values of functions in weighted Hardy spaces

Let $0<p<\mathrm{\infty }$ and $0\le q<\mathrm{\infty }$. For each $f$ in the weighted Hardy space $H_{p,q}$,\ we show that $d\mathrm{\parallel }{f}_r{\mathrm{\parallel }}_{p,q}^p\mathrm{/}dr$ grows at most like $o(1\mathrm{/}1-r)$ as $r\to 1$.