We prove an optimal Kawamata-Miyaoka-type inequality for terminal $\mathbb Q$-Fano threefolds with Fano index at least $3$. As an application, any terminal $\mathbb Q$-Fano threefold $X$ satisfies the following Kawamata-Miyaoka-type inequality \[ c_1(X)^3 < 3c_2(X)c_1(X). \]
Comment: 33 pages, 4 tables. Any comments are welcome. v2: we improve the exposition, 29 pages, 3 tables. v3: Final published vesion