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On integer-valued entire functions at the Gaussian integers satisfying additional congruence conditions. (English) Zbl 1161.30014

The author extends the result of Gramain on integer-valued entire functions at the Gaussian integers to functions that together with their first \(s-1\) derivatives take integral values and in addition satisfy certain congruence conditions. More precisely, the following theorem is proved:
Theorem. Let \(s\) be a positive integer and \(f\) be an entire transcendental function, that is not a polynomial function, such that for all \(\sigma = 0, \ldots, s-1 \), the following two conditions are satisfied:
(a) \(f^{(\sigma)}(\mathbb{Z}[i])\subset \mathbb{Z}[i],\)
(b) for all \(\zeta \in \mathbb{Z}[i]\) and all Gaussian primes \( p\in \mathbb{Z}[i], \) we have \[ f^{(\sigma )}(\zeta+p)-f^{(\sigma )}(\zeta )\in p\mathbb{Z}[i]. \]
Then \[ \limsup_{r \to \infty}\frac{\log |f|_r}{r^2}\geq \frac{\pi s }{2e}e^{1/\pi s}. \]
The author also shows that without condition (b) the following estimate is true \[ \limsup_{r \to \infty}\frac{\log |f|_r}{r^2}\geq \frac{\pi s }{2e}. \]

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
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