In 1959 Hayman proved an inequality from which it follows that if/ is transcendental and meromorphic in the plane then either/ takes every finite complex value infinitely often or each derivative / ( k ) , k > 1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account.2000 Mathematics subject classification: primary 3OD35. IntroductionOur starting point is the following result of Hayman [8,9]. THEOREM 1.1. Let k e H and let a, b e C, b ^ 0. Let f be nonconstant and meromorphic in the plane. Then at least one of f -a and f ( A r ) -b has at least one zero, and infinitely many iff is transcendental.Hayman's proof of Theorem 1.1 is based on Nevanlinna theory, the result deduced from the following inequality. Here the notation is that of [9], with S(r,f) denoting any term which is o (T(r,f)) as r -> oo, possibly outside a set of finite Lebesgue measure. (1) r(rJ)