The behaviour on the rays of functions from the Bergman and Fock spaces (Q1025439)

It follows from the classical Fejér-Riesz inequality that \(\int_0^1|f(t)|^2dt<\infty\) whenever \(f\in H^2\). The author establishes similar results for the Bergman space \(A^{2,s}\) of all functions, \(f\), holomorphic in the unit disk \({\mathbb D}\) for which \[ \int_{\mathbb D}|f(z)|^2 (1-|z|^2)^s dm_2(z)<\infty, s>-1 \] and the Fock space \(\mathcal F_s\) of all entire functions \(f\) such that \[ \int_{\mathbb C}|f(z)|^2 e^{-s|z|^2}dm_2(z)<\infty, s>0. \] It is shown that \(I(s):=\int_0^1 |f(t)|^2(1-t^2)^{s+1}dt<\infty\) whenever \(f\in A^{2,s}\) and \(J(s):=\int_{0}^\infty |f(t)|^2 t e^{-st^2}dt<\infty\) whenever \(f\in \mathcal F_s\). It is also shown that these results are best possible in the sense that \(s\) cannot be replaced by \(\sigma<s\) in \(I(s)\) or \(J(s)\).