On the asymptotic convergence of sequences of analytic functions (Q2474130)

In this article, sequences \(\{ f_{n}\}\) of analytic self-mappings of a domain \(\Omega\subset\mathbb{C}\) are considered, together with the associated sequence \(\{\Theta_{n}\}\) of inner compositions given by \(\Theta_{n}=f_{1}\circ f_{2}\circ\cdots\circ f_{n}\). A sequence \(\{ f_{n}\}\) converges asymptotically to a function \(f\), if for any sequence of integers \(\{ n_{k}\}\) with \(n_{1}<n_{2}<\cdots\), \(\lim_{k\rightarrow\infty}(f_{n_{k}}\circ f_{n_{k}+1}\circ\cdots\circ f_{n_{k+1}-1}-f^{n_{k+1}-n_{k}})=0\) locally uniformly in \(\Omega\). The key results obtained in this paper are as follows: (1) Let \(\{ f_{n}\}\) be a sequence of univalent self-mappings of the unit disc \(D\). If \(\{ \Theta_{n}\}\) converges in \(D\), as \(n\rightarrow\infty\), to a non-constant limit function \(\Theta\), then \(\{ f_{n}\}\) converges asymptotically to the identity in \(D\), and, in particular, \(\{ f_{n}\}\) converges locally uniformly to the identity in \(D\). Moreover, for every \(m\geq 1\), \(\Theta =f_{1}\circ f_{2}\circ\cdots\circ f_{m-1}\circ\lim_{n\rightarrow\infty}(f_{m}\circ f_{m+1}\circ\cdots\circ f_{n})\), where \(\lim_{n\rightarrow\infty}(f_{m}\circ f_{m+1}\circ\cdots\circ f_{n})\) exists and is non-constant. (2) Let \(f_{n}:\Omega\rightarrow\mathbb{C}\) be a sequence of analytic functions that converges asymptotically to the identity in \(\Omega\). If the sequence \(\{ \Theta_{n}\}\) is a normal family in \(\Omega\), then either \(\{ \Theta_{n}\}\) converges locally uniformly in \(\Omega\) to an analytic function \(\Theta\), or \(\{ \Theta_{n}\}\rightarrow\infty\) in \(\Omega\). In addition, further considerations are made to the case where a sequence \(\{ f_{n}\}\) of entire functions converges uniformly asymptotically to the identity function in \(\mathbb{C}\).