  In this work we consider compact K\"ahler manifolds with non-positive mixed
curvature which is a "convex combination" of Ricci curvature and holomorphic
sectional curvature. We show that in this case, the canonical line bundle is
nef. Moreover, if the curvature is negative at some point, then the manifold is
projective with canonical line bundle being big and nef. If in addition the
curvature is negative, then the canonical line bundle is ample. As an
application, we answer a question of Ni concerning manifolds with negative
$k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapani to
the conformally K\"ahler case. We also show that the compact K\"ahler manifold
is projective and simply connected if the mixed curvature is positive.
