\(Q_K\) type spaces of analytic functions (Q2369472)

The authors introduce a family of function spaces called \(Q_K(p,q).\) The space \newline \(Q_K(p,q)\) consists of all functions \(f\) holomorphic in the unit disk \(\Delta\) such that \[ \sup_{a\in \Delta} \int_{\Delta} | f'(z)| ^p (1-| z| )^qK(g(z,a))dA(z) < \infty, \] where \(p > 0,\;q> -2,\;\) \(g\) is the Green function of \(\Delta\) and \(K\) is an increasing function from \([0,\infty)\) into \([0,\infty).\) For particular choices of \(K,p\) and \(q\) these spaces have been previously studied by Essén and Wulan, R.Zhao and others. In the paper under review the authors prove that \(Q_K(p,q)\) is always contained in the ``alpha Bloch space''\({\mathcal B}^{\alpha}\) with \(\alpha = \frac{q+2}{p},\) and that the two spaces coincide if and only if a certain integral involving \(K\) converges. They also prove some related results.