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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
##### Keywords:
universal Taylor series; boundary behaviour
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##### References:
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