A family of transcendental meromorphic functions, fp(z), p ∈ N is considered. It is shown that, if p [ges ] 6, then the Hausdorff dimension of the Julia set of λfp satisfies dim J(λfp) [les ] 1/p, for 0 < λ < 1/6p, and dim J(λfp) [ges ] 1−(30 ln ln p/ln p), for p4p−1/105 ln p < λ < p4p−1/104 ln p. These results are used elsewhere to show that, for each d ∈ (0, 1), there exists a transcendental meromorphic function for which dim J(f) = d.