The partition function for ferromagnetic plane rotators in a complex external field , with ¦Im ¦ ¦Re ¦, is bounded below in modulus by its value at =0. The proof is based on complex combinations of duplicated variables which are positive definite on a subgroup of the configuration group. In the isotropic situation (and =0), the associated Gaussian inequalities imply that all truncated correlation functions decay at least as the two-point function.