Dimensions of Julia sets of meromorphic functions with finitely many poles

Let $f$ be a transcendental meromorphic function with finitely many poles such that the finite singularities of $f^{-1}$ lie in a bounded set. We show that the Julia set of $f$ has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions