A generic result concerning univalent universal functions
type: Item
instance of: scholarly article
title: A generic result concerning univalent universal functions
zbMATH Open document ID: 1060.30007
published in: Archiv der Mathematik
publication date: 2004-10-05T00:00:00Z
review text: Assume that \(G\) is a Jordan domain in the complex plane \(\mathbb C\), and that \(K \subset \mathbb C\) is a compact set. By \(H(G)\), \(A(G)\), \(A(K)\), \(B(G)\) it is denoted, respectively, the space of all holomorphic functions on \(G\), the space of continuous functions on \(\overline{G}\) that are holomorphic in \(G\), the space of continuous functions on \(K\) that are holomorphic in \(K^0\), and the subspace of \(A(G)\) consisting of functions that are either univalent or constant on \(G\). If \(A(G)\) is endowed with the maximum norm, then it becomes a Banach space and \(B(G)\) is closed in it, so \(B(G)\) is a Baire space. In addition, if \(a \in G\) then \(W\) denotes the class of holomorphic functions that are universal under derivatives and universal with respect to overconvergence, and performing both approximations with the same approximative sequence, that is, \(f \in W\) whenever \(f \in H(G)\) and, for every choice of compact sets \(L \subset G\) and \(K \subset \overline{G}^c\) with \(L^c\) and \(K^c\) connected and for every choice of functions \(\phi \in A(K)\), \(h \in A(L)\), there exists a sequence \((n_j)\) of natural numbers such that \(\lim_{j \to \infty} \sup_{z \in K}| S_{n_j}(f,a)(z) - \phi (z)| = 0 = \lim_{j \to \infty} \sup_{z \in L} | f^{(n_j)}(z) - h(z)| \), where \(S_n(f,a)\) is the \(n\)th-partial sum of the Taylor expansion of \(f\) with center \(a\). By using a modified version of a classical result of J.~Walsh concerning approximation of holomorphic functions by polynomials, the authors prove that \(W \cap B(G)\) is residual in \(B(G)\). Hence there exist functions in \(A(G)\) that are universal (in both senses of derivation and overconvergence) and also univalent in \(G\).
reviewed by: L. Bernal-González
Mathematics Subject Classification ID: 30B30
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 2106409
zbMATH Keywords: univalent holomorphic function
zbMATH Keywords: universal under derivatives
zbMATH Keywords: universal with respect to overconvergence
author: G. Costakis
author: V. Vlachou
MaRDI profile type: Publication
full work available at URL: s00013-003-0600-z
OpenAlex ID: W2061866139
DOI: 10.1007/S00013-003-0600-Z
description: scientific article; zbMATH DE number 2106409