Gabriel's and Frazer's problems for weighted Bergman spaces and their applications

In this paper, we investigate Gabriel's and Frazer's problems for analytic and complex-valued harmonic weighted Bergman spaces. More precisely, we establish weighted integral inequalities of the form
\[
\int_C |f(z)|^p(1-|z|^2)^{\alpha+1}\,|dz|
\leq
K_{p,\alpha,C}
\int_{\mathbb D}|f(z)|^p(1-|z|^2)^\alpha\,dA(z),
\]
where $f$ is an analytic or complex-valued harmonic function on the unit disk $\mathbb D$ and $C$ is an arbitrary convex curve contained in $\mathbb{D}$. The corresponding problem was first studied by Gabriel [Proc. Lond. Math. Soc. 28 (1928), 121--127] for analytic Hardy spaces, where the inequality holds for every $0<p<\infty$. In contrast, the harmonic Hardy space analogue was recently shown to fail whenever $0<p\le1$. We prove that this phenomenon does not occur in the weighted harmonic Bergman setting by establishing Gabriel's inequality throughout the full range $0<p<\infty$. We further study Frazer's problem for circles and for the union of two intersecting diameters. As important applications of our main results, we derive Gabriel-type and Frazer-type inequalities for the analytic and harmonic Möbius invariant spaces $Q(n,p,\alpha)$ and $Q_h(n,p,\alpha)$.