On transcendental analytic functions mapping an uncountable class of $U$-numbers into Liouville numbers

In this paper, we shall prove, for any $m\ge 1$, the existence of an uncountable subset of $U$-numbers of type $\le m$ (which we called the set of {\it $m$-ultra numbers}) for which there exists uncountably many transcendental analytic functions mapping it into Liouville numbers.