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

Chengji Xiong, Junming Liu

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