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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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