On the Hausdorff dimension of the escaping set of certain meromorphic functions

Let f f be a transcendental meromorphic function of finite order ρ \rho for which the set of finite singularities of f − 1 f^{-1} is bounded. Suppose that ∞ \infty is not an asymptotic value and that there exists M ∈ N M \in \mathbb N such that the multiplicity of all poles, except possibly finitely many, is at most M M . For R > 0 R>0 let I R ( f ) I_R(f) be the set of all z ∈ C z\in \mathbb {C} for which lim inf n → ∞ | f n ( z ) | ≥ R \liminf _{n\to \infty }|f^n(z)|\geq R as n → ∞ n\to \infty . Here