An upper bound for the GSV-index of a foliation

Let F be a holomorphic foliation at p∈C2, and let B be a separatrix of F. We prove the following upper bound GSVp(F,B)≤4τp(F,B)-3μp(F,B), where GSVp(F,B) is the Gómez-Mont-Seade-Verjovsky index of the foliation F with respect to B, μp(F,B) is the multiplicity of F along B and τp(F,B) is the dimension of the quotient of C{x,y} by the ideal generated by the components of any 1-form defining F and any equation of B.