On the zeros of functions from Bergman spaces and some related spaces (Q650271)

Denote by \(A_{q,\alpha}^p\) the space of all functions \(f\) analytic in the unit disk such that \[ \left(\int_0^1 M_p(f;r)^q (1-r^2)^{\alpha} r dr\right)^{1/q}<\infty, \] where \(0<p\leq \infty\), \(0<q<\infty\) and \(-1 < \alpha<\infty\). \(M_{\infty}(f;r)\) denotes the standard maximum modulus; for \(p<\infty\), \[ M_{p}(f;r):=\left(\frac{1}{2\pi}\int_0^{2\pi} | f(r e^{i\theta})| ^p d\theta\right)^{1/p}. \] (Setting \(p=q\) yields the Bergman space with standard weight \(t^{\alpha}\).) Set \[ \displaystyle A_{q,\alpha}^0 =\bigcup_{p>0} A_{q,\alpha}^p. \] The article mainly deals with estimates of the zero-counting function \(n(r;f)\). Inspired by certain results of Sedletskii the author proves the following: Let \(f\in A_{q,\alpha}^0\). Then for every \(r\) sufficiently close to \(1\), \[ n(r;f)<\frac{\alpha +1}{q} \cdot\frac{1}{1-r} \cdot\left(\log \frac{\varepsilon_f(r)}{1-r} + \log \log \frac{\varepsilon_f(r)}{1-r} \right), \] where \(\varepsilon_f(r)\to 0\) as \(r\to 1\), and \(\varepsilon_f(r)\) cannot be replaced by any standard infinitesimal. The author also considers the behaviour of \(1-| z_n(f)|\) for \(n\to\infty\), where \(\{z_n(f)\}\) denotes the set of zeros of \(f\).