We prove a Miyaoka–Yau type inequality for threefolds such that −KX is nef and of numerical dimension ≥ 2 \(A \cdot \left(c_{2}(X)+\lambda{c}_{1}^{2}(X)\right) >0,\) where A is an ample line bundle and λ > 0 is a constant dependent on the characteristic p and the Cartier index n0 of KX. This is an analogue of the Miyaoka–Yau type inequality presented in [Algebra Number Theory, 2022, 16(10): 2339–2384], which treats minimal threefolds of general type. We attain this inequality by following the strategy of [Algebra Number Theory, 2022, 16(10): 2339–2384, Duke Math. J., 2019, 168(7): 1269–1301]. Applying a similar argument, we also show that in characteristic ≥ 5, if −KX is nef and −KX · c2(X) < 0, then the nonvanishing theorem for anti-canonical divisor holds.
