Holomorphic sectional curvature, nefness and Miyaoka-Yau type inequality

On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a K\"ahler metric with semi-negative holomorphic sectional curvature. We prove that a compact K\"ahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed K\"ahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.