## 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