Let f be a nonconstant, nonlinear meromorphic function. We study the (non-)existence of invariant line fields on the Julia set of such a function under certain nonrecurrence conditions.

More precisely, suppose that the finite part of the postsingular set of f is bounded, and that f has no recurrent critical points, no wandering domains, and no pre-poles of arbitrarily high order. Then we show that f supports no invariant line fields on its Julia set.

This is obtained essentially from two separate results: the absence of invariant line fields for "measurably transitive" meromorphic functions (which in the rational case is due to McMullen) and a version of Mane's theorem for rational functions. Both of these (and our main theorem) extend results of Graczyk, Kotus and Świątek.