Baker domains of meromorphic functions

Let $f$ be a transcendental meromorphic function and $U$ be an invariant Baker domain of $f$. We obtain a new estimate for the growth of the iterates of $f$ in $U$, and we use this estimate to improve an earlier result relating the geometric properties of $U$ and the proximity of $f$ in $U$ to the identity function. We illustrate the latter result by considering transcendental meromorphic functions $f$ of the form $ f(z) = az + bz^ke^{-z}(1+o(1)) \; \mbox{ as } \Re (z) \rightarrow \infty, $ where $k \in \bf N$, $a > 1$ and $b > 0$, and we show that these functions have Baker domains which contain an unbounded set of critical points and an unbounded set of critical values