Angular limits and infinite asymptotic values of analytic functions of slow growth (Q913985)

If \(w=f(z)\) is defined in the unit disk \(\Delta\) : \(| z| <1\), then the set A(f,w) consists of those points \(\zeta\) on the unit circle C: \(| z| =1\) at which f has asymptotic value w; \(A(f)=\cup_{w}A(f,w)\). The class \({\mathcal A}\) of G. R. MacLane is the set of all nonconstant analytic functions f on \(\Delta\) for which \(A(f)\) is dense on C. MacLane proved that if \(f\in {\mathcal A}\) and \(A(f,\infty)\cap J=\emptyset\) for an open subarc J of C, then \(| A(f)\cap J| >0\). A sufficient condition for \(f\in {\mathcal A}\) is \[ \int^{1}_{0} \log^+ \log^+ M(f;r)dr<+\infty. \] The author considers related notions. An admissible growth function \(\nu\) is any increasing, unbounded, continuous function \(\nu\) : [0,1)\(\to (1,+\infty)\). The class \({\mathcal A}_{\nu}\) consists of all nonconstant analytic functions f on \(\Delta\) such that M(f;r)\(\leq \nu (r)\). If \(\omega\) (t), \(t>0\), is an increasing, continuous, concave-downward function with \(\omega (0)=0\), \(\omega '(0)=+\infty\), then \(\nu (r)=\nu_{\omega}(r)=\exp [\omega (1- r)/(1-r)]\), \(0<r<1\), is an admissible growth function. Associated with \(\omega\) is also the usual Hausdorff measure \(H_{\omega}.\) Theorem 1. Let \(f\in {\mathcal A}_{\nu}\), where \(\nu =\nu_{\omega}\). If \(H_{\omega}(A(f,\infty)\cap J)=0\) for an open subarc J of C, then \(| F(f)\cap J| >0\), where F(f) is the set of Fatou points of f. Theorem 2. For any admissible growth function \(\nu\), there exists a function \(f\in {\mathcal A}_{\nu}\) such that \(| A(f)| =0\).