proper lk-biharmonic hypersurfaces in the euclidean sphere with two principal curvatures

In this paper we classify proper lk-biharmonic hypersurfaces m, in the unit euclidean sphere should have two principal curvatures and we show that they are open pieces of standard products of spheres. also we study proper lk- biharmonic compact hypersurfacesm with respect to tr(s² o pk) and hk where s is the shape operator, pk is the newton transformation and hk is the k-th mean curvature ofm, and by definiteness assumption of pk, we show that hk+1 is constant.