Gabriel's problem for harmonic Hardy spaces

Abstract

We obtain inequalities of the form$\underset{C}{\int }\left|f\right(z\left){|}^p\right|dz|\le A(p\left)\underset{T}{\int }\right|f\left(z\right){|}^p\left|dz\right|(p>1),$ where f is harmonic in the unit disk $D$, $T$ is the unit circle, and C is any convex curve in $D$. Such inequalities were originally studied for analytic functions by R.M. Gabriel (1928) [11]. We show that, unlike in the case of analytic functions, such inequalities cannot be true in general for $0<p\le 1$. Therefore, we produce an inequality of a slightly different type, which deals with the case $0<p<1$. An example is given to show that this result is “best possible”, in the sense that an extension to $p=1$ fails. Then we consider the special case when C is a circle, and prove a refined result that curiously holds for $p=1$ as well. We conclude with a maximal theorem which has potential applications.