  In this paper, we study $4$-dimensional complete hypersurfaces with
$w$-constant mean curvature in the unit sphere. We give a lower bound of the
scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant
mean curvature. As a by-product, we give a new proof of the result of
Deng-Gu-Wei under the weaker topological condition.
