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On the growth of universal functions. (English) Zbl 0973.30002
Let \(f=\sum_{k=0}^\infty a_kz^k\) be holomorphic in the unit disc \(D\). Put \(S_n(f):=\sum_{k=0}^na_kz^k\). Assume \(f\) is universal, i.e. for any compact set \(K\subset\mathbb C\setminus D\) for which \(\mathbb C\setminus K\) is connected, and for any function \(h\in\mathcal C(K)\cap\mathcal O(\text{int}K)\), there exists a sequence \((n_k)_{k=1}^\infty\) such that \(S_{n_k}(f)\longrightarrow h\) uniformly on \(K\). The author proves that the growth of \(f\) cannot be of the type \(|f(z)|\leq C\exp(\varPhi(\frac 1{1-|z|}))\), where \(\varPhi:[1,+\infty)\longrightarrow[1,+\infty)\) is a continuous increasing function with \(\int_1^\infty\log\varPhi(t)\frac{dt}{t^2}<+\infty\). On the other hand, he constructs a universal function \(F\) of the growth \(|F(z)|\leq C\exp(\exp(\frac{M}{1-|z|}\log\log\frac 4{1-|z|}))\), where \(M\) is a positive constant. The author also studies the value distribution of \(f\). In particular, he shows that for every \(a\in\mathbb C\) except at most one value, the equation \(f(z)=a\) has an infinite number of distinct solutions \((z_j^\ast(a))_{j=1}^\infty\) such that \(\sum_{j=1}^\infty h(1-|z_j^\ast(a)|)=+\infty\) for any increasing continuous function \(h:(0,1]\rightarrow(0,+\infty)\) with \(\int_0^1\log\frac 1{h(s)}ds<+\infty\).

MSC:
30B10 Power series (including lacunary series) in one complex variable
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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