  We derive some integral inequalities for holomorphic maps between complex
manifolds. As applications, some rigidity and degeneracy theorems for
holomorphic maps without assuming any pointwise curvature signs for both the
domain and target manifolds are proved, in which key roles are played by total
integration of the function of the first eigenvalue of second Ricci curvature
and an almost nonpositivity notion for holomorphic sectional curvature
introduced in our previous work. We also apply these integral inequalities to
discuss the infinite-time singularity type of the K\"ahler-Ricci flow. The
equality case is characterized for some special settings.
