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Boundary behaviour of functions which possess universal Taylor series. (English) Zbl 1272.30081
Let \(\Omega\) be a simply connected proper subdomain of the complex plane \(\mathbb{C}\) and \(\zeta \in \Omega\). A holomorphic function \(f\) on \(\Omega\) possesses a universal Taylor series expansion about \(\zeta\), for short \(f \in {\mathcal U}(\Omega,\zeta)\), if the partial sums \[ S_N(f,\zeta) = \sum_{n=0}^N \frac{f^{(n)}(\zeta)}{n!} (z-\zeta)^n \] satisfy the following property: For every compact set \(K \subset \mathbb{C}\setminus \Omega\) with connected complement and every function \(g\) which is continuous on \(K\) and holomorphic in the interior of \(K\), there is a subsequence \(\big( S_{N_k}(f,\zeta)\big)\) that converges to \(g\) uniformly on \(K\).
The main result of the present paper, proven by an extensive use of potential theoretic tools, is: Given \(f \in {\mathcal U}(\Omega,\zeta)\), then, for any boundary point \(w \in \partial \Omega\), any \(r > 0\) and any component \(U\) of \(\{ z \in \Omega : |z-w| < r\}\), the function \(\log^+ |f|\) does not have a harmonic majorant on \(U\), and thus \(\mathbb{C} \setminus f(U)\) is polar. As a corollory, \(f\) is unbounded near every point of \(\partial \Omega\).
This theorem contains or generalises earlier results by A. Melas et al. [J. Anal. Math. 73, 187–202 (1997; Zbl 0894.30003)], as well as F. Bayart [Rev. Mat. Complut. 19, No. 1, 235–247 (2006; Zbl 1103.30003)].

MSC:
30K05 Universal Taylor series in one complex variable
30B30 Boundary behavior of power series in one complex variable; over-convergence
30E10 Approximation in the complex plane
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