  E. Maillet proved that the set of Liouville numbers is preserved under
rational functions with rational coefficients. Based on this result, a problem
posed by Kurt Mahler is to investigate whether there exist entire
transcendental functions with this property or not. For large parametrized
classes of Liouville numbers, we construct such functions and moreover we show
that it can be constructed such that all their derivatives share this property.
We use a completely different approach than in a recent paper, where functions
with a different invariant subclass of Liouville numbers were constructed
(though with no information on derivatives). More generally, we study the image
of Liouville numbers under analytic functions, with particular attention to
$f(z)=z^{q}$ where $q$ is a rational number.
