Gabriel's problem for harmonic Hardy spaces

We obtain inequalities of the form ∫C |f(z)|ᵖ |dz| ≤ A(p) ∫T |f(z)|ᵖ |dz|, (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 [Proc. London Math. Soc. 28(2), 1928]. We show that these results, unlike in the case of analytic functions, cannot be true in general for 0< p ≤ 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 which surprisingly holds for $p=1$ as well. We conclude with a maximal theorem which has potential applications.