A Note On Transcendental Analytic Functions With Rational Coefficients Mapping $\mathbb{Q}$ Into Itself

In this note, the main focus is on a question about transcendental entire functions mapping $\mathbb{Q}$ into $\mathbb{Q}$ (which is related to a Mahler’s problem). In particular, we prove that, for any $t>0$, there is no a transcendental entire function $f\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and whose denominator of $f(p\mathrm{/}q)$ is $O(q^t)$, for all rational numbers $p\mathrm{/}q$, with $q$ sufficiently large.