We show analyticity of the pressure for some classical ferromagnetic systems in the region ¦Im exp(−S6n−S4n+np=12pS2p) , where 2n is an arbitrary real number and the other parameters are positive. It also applies directly to plane rotators in the region ¦Im ¦ ¦Re ¦ (Euclidean norms), but the proof will be given in a subsequent article, together with new inequalities between truncated correlation functions.