Interpolation Banach lattices containing no isomorphic copies of c0

This article is concerned with the question of when the Banach lattices generated by the interpolation method of constants contain no isomorphic copies of c0. For this we study the Köthe dual spaces of Banach lattices obtained by the methods of constants and means. We find an explicit formula for the Köthe dual of Banach lattices generated by the means method with a mild restriction on the parameters. We apply these results to describe a large class of Banach lattices, generated by the method of constants, containing no subspaces isomorphic to c0. Moreover, we present applications to the strict singularity of certain inclusion maps between Banach sequence lattices.