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 $${\int }_C\mathrm{\mid }f(z){\mathrm{\mid }}^p(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{\alpha +1}\mathrm{\mid }dz\mathrm{\mid }\le K_{p,\alpha ,C}{\int }_{\mathbb{D}}\mathrm{\mid }f(z){\mathrm{\mid }}^p(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{\alpha }dA(z),$$ 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 $0<p<\mathrm{\infty }$. In contrast, the harmonic Hardy space analogue was recently shown to fail whenever $0<p\le 1$. We prove that this phenomenon does not occur in the weighted harmonic Bergman setting by establishing Gabriel’s inequality throughout the full range $0<p<\mathrm{\infty }$. 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 $Q_h(n,p,\alpha )$.