A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

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