  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 $f(p/q)$ is
$O(q^{t})$, for all rational numbers $p/q$, with $q$ sufficiently large.
