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

TL;DR

This paper investigates the growth rate of the mean values of functions in weighted Hardy spaces, establishing an upper bound on how quickly these mean values can increase as the boundary is approached.

Contribution

It provides a new result on the asymptotic behavior of the derivatives of mean values in weighted Hardy spaces, extending understanding of boundary growth in these function spaces.

Findings

The derivative of the mean value norm grows slower than 1/(1-r) as r approaches 1.

The growth rate is at most o(1/1-r), indicating a controlled increase.

Results apply to functions in weighted Hardy spaces with parameters 0<p<∞ and 0≤q<∞.

Abstract

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