Limiting behaviour of functions of locally bounded characteristic (Q891358)

Let \(f\) be a meromorphic function in \(|z|<R\). If \(f\) satisfies \(T(R)=\lim_{r\to R-}T(r)<\infty\), we say that \(f\) has bounded characteristic (b.c.) in \(|z|<R\). Denote by \(D_0\) the unit disk \(|z|<1\). A subset \(F_0\) of \(D_0\) is called an \(F\)-set if it is contained in a union \(\mathcal F\) of disks \(D(z_0,\rho)\) lying in \(D_0\) and with the following properties. Define \(r_k=1-2^{-k}\), \(k=0, 1, 2,\dots\), and let \(\rho_k\) be the sum of the radii of all the disks in \(\mathcal F\) whose centers \(z_0\) satisfy \(r_{k-1}\leq |z_0|<r_k\). If, for some positive integer \(k_0\), \(\rho_k\) is finite for \(k\geq k_0\) and \(\sum_{k=k_0}^\infty \big(\log(2^{1-k}/\rho_k)\big)^{-1}<\infty\), then \(F_0\) is called an \(F\)-set. In [Bull. Lond. Math. Soc. 46, No. 3, 537--549 (2014; Zbl 1295.30120)], the author obtained the limiting behaviour of functions of b.c. in \(D_0\). Suppose that \(f\) is meromorphic of b.c. in \(D_0\). Then there exists an \(F\)-set \(F_0\) such that if \(\zeta=e^{i\theta}\) \[ \frac{|\zeta-z|^2}{1-|z|^2}\log |f(z)|\to \alpha(\theta,f)\tag{1} \] as \(z\to e^{i\theta}\) nontangentially from \(D_0\) with \(z\not\in F_0\). Here \(-\infty<\alpha(\theta,f)<\infty\) and \(\alpha(\theta,f)=0\) outside a countable set \(\{\theta_\nu\}\), where \[ \sum |\alpha(\theta_\nu,f)|<\infty.\tag{2} \] The author proves a partial extension of this result for locally bounded characteristic (l.b.c.) defined below. Consider a function \(\varphi(z)\) which is analytic in \(D_0\) and maps \(D_0\) into itself. If \(\varphi\) is real and increasing in \([0,1]\), \(\varphi(0)=r_0\geq0\), \(\varphi(1)=1\) and \(1-\varphi(z)<C_1(1-z)\), \(0<z<1\), \(\varphi\) is called a support function, where \(C_1\) is a constant. If \(f\) is meromorphic in \(D_0\), and if there exists a support function \(\varphi\) such that \(T(r,f(e^{i\theta}\varphi(z)))\leq C_2\), \(|\theta|\leq \pi\), \(0<r<1\), then we say that \(f\) has l.b.c., where \(C_2\) is a constant independent of a parameter \(\theta\). It is shown in this paper that, if \(f\) is a meromorphic function having l.b.c. in \(D_0\), then ({1}) still holds for every boundary point \(\zeta=e^{i\theta}\) of \(D_0\), but the exceptional set \(F_0\) may now depend on \(\zeta\). It is still true that \(\alpha(\theta,f)=0\) outside a countable set \(\{\theta_\nu\}\) and that \(|\alpha(\theta_\nu,f)|\) is bounded. Further, its derivative \(f'\), and hence all its derivatives have l.b.c., and \(\alpha(\theta,f')\leq \alpha(\theta,f)\), \(|\theta|\leq \pi\), with equality if \(\alpha(\theta,f)\neq0\). The author gives a comment that it is not known whether ({2}) holds or even whether \(\alpha(\theta_\nu,f)\to0\) as \(\nu\to\infty\).