The discovery of the Higgs particle at 125 GeV is demanding a detailed knowledge of the properties of this fundamental component of the Standard Model. Various proposals of electron and muon colliders have been put forward for precision studies of the partial widths of the various decay channels. It will be shown that in the case of a Higgs factory through a muon collider, sizeable radiative effects -of order of 50% - must be carefully taken into account for a precise measurement of the leptonic and total widths of the Higgsparticle. Similar effects do not apply in the case of Higgs production inelectron-positron colliders.