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Boundary behaviour of universal Taylor series. (Comportement à la frontière des séries de Taylor universelles.) (English. French summary) Zbl 1297.30083
The authors improve several known results on the strong growth properties at every boundary point of functions in \(\mathcal{U}\), the set of all universal Taylor series, in the literature also denoted as \(\mathcal{U}(\mathbb{D},0)\). A power series \(f\) that converges on the unit disc \(\mathbb{D}\) belongs to \(\mathcal{U}\) if its partial sums approximate arbitrary polynomials on arbitrary compacta in \(\mathbb{C} \setminus \mathbb{D}\) that have connected complement.
In detail they show that if \(\psi : [0,1) \to (0,\infty)\) is an increasing function with \(\int_0^1 \log^+ \log^+ \psi(t) \,dt < \infty\), and the power series \(f\) satisfies \(|f(z)| \leq \psi(|z|)\) on \(D(w,r) \cap \mathbb{D}\), \(D(w,r) = \{ z : |z-w| < r\}\), for some \(w\) on the unit circle \(\mathbb{T}\) and \(r > 0\), then \(f \notin U\). An analogue of this theorem holds for universal polynomial expansions of harmonic functions in terms of homogeneous polynomials.
As a corollary they obtain a Picard-type property of universal Taylor series. A function \(f \in \mathcal{U}\) assumes every complex value, with at most one exception, infinitely often on \(D(w,r) \cap \mathbb{D}\) for every \(w \in \mathbb{T}\) and \(r > 0\).
They further prove that any function \(f \in \mathcal{U}\) must assume all but one complex value in any angle at “most” boundary points. As angular approach regions they consider \[ \Gamma_\alpha^t(w) := \big\{ z : 1-t < |z| < 1, |z-w| < \alpha\big( 1 - |z| \big) \big\}, \quad w \in \mathbb{T}, \alpha > 1, t \in (0,1]. \] “Most” points means that \(\mathbb C \setminus f\big( \Gamma_\alpha^t(w) \big)\) contains at most one point for all \(w\) in a residual set \(E \subset \mathbb{T}\). Moreover, no member of \(\mathcal{U}\) belongs to any Bergman or Bergman-Nevanlinna class on \(\mathbb{D}\).

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