ON THE HAUSDORFF DIMENSION OF THE ESCAPING SET OF CERTAIN MEROMORPHIC FUNCTIONS

Let f be a transcendental meromorphic function of finite order rho for which the set of finite singularities of f(-1) is bounded. Suppose that infinity is not an asymptotic value and that there exists M is an element of N such that the multiplicity of all poles, except possibly finitely many, is at most M. For R > 0 let I-R(f) be the set of all z is an element of C for which lim If(n ->infinity)vertical bar f(n)(z)vertical bar >= R as n -> infinity. Here f(n) denotes the n-th iterate of f. Let I(f) be the set of all z is an element of C such that vertical bar f(n)(z)vertical bar -> infinity as n -> infinity; that is, I(f) = boolean AND I-R>0(R)(f). Denote the Hausdorff dimension of a set A subset of C by HD(A). It is shown that lim(R ->infinity) HD(I-R(f)) <= 2M rho/(2 + M rho). In particular, HD(I(f)) <= 2M rho/(2 + M rho). These estimates are best possible: for given rho and M we construct a function f such that HD(I(f)) = 2M rho/(2 + M rho) and HD (I-R(f)) > 2M rho/(2 + M rho) for all R > 0. If f is as above but of infinite order, then the area of I-R(f) is zero. This result does not hold without a restriction on the multiplicity of the poles.