In the paper Curvature estimates for minimal hypersurfaces written by R.Schoen, L.Simon and S.T.Yau, in the middle of page 279, there is a inequality
$2|K_{n+1,iji}|\leq K_1-K_2$
where $K_{n+1,iji}$is the curvature of a Riemannian manifold $N$ and $K_1$ and $K_2$ are upper bound and lower bound of sectional curvature of $N$ respectively.
I don't know how this inequality is obtained.