Baker domains of meromorphic functions

Abstract

Let $f$ be a transcendental meromorphic function and $U$ a Baker domain of $f$. We obtain new estimates for the behaviour of the iterates of $f$ in $U$ and we use these estimates to improve an earlier result relating the geometric properties of $U$ to the proximity of $f$ to the identity function in $U$. We also apply these estimates to Baker domains of transcendental meromorphic functions $f$ of the form

\begin{gather*} f(z) = az + bz^ke^{-z}(1+o(1)) \quad \text{as } \Re (z) \rightarrow \infty, \end{gather*}

where $k \in {\mathbb N},\ a > 1$ and $b > 0$, and show that these Baker domains contain an unbounded set of critical points and an unbounded set of critical values.