Area inequality and \(Q_{p}\) norm
Let \(R\) be a hyperbolic Riemann surface, and let \(f\) be a function defined and analytic on \(R\). The main result of the paper under review is the following theorem: for any \(p>q\geq 0\) and any \(a\in R\), we have \[ \int\int_R | f'(z)| ^2 g^p(z,a)\,dx\,dy \leq \frac{2^{q-p}\Gamma(p+1)}{\Gamma(q+1)} \int\int_R | f'(z)| ^2 g^q(z,a)\,dx\,dy, \] where \(g(z,a)\) denotes a Green's function of \(R\) with logarithmic singularity at \(a\). The theorem is a generalization of results of [\textit{S. Kobayashi}, Can. J. Math. 36, 747--755 (1984; Zbl 0553.30022)] and of [\textit{T. A. Metzger}, Can. J. Math. 33, 1255--1260 (1981; Zbl 0442.30030)].
scientific article; zbMATH DE number 2160152