Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds.