It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can find the following

Theorem 1. Assume that $p = \mathrm{char}(k) > 0$. Let $X$ be a $d$-dimensional nonsingular projective $k$-variety with an ample line bundle $H$. Let $E$ be a $p$-semistable (with respect to $H$) vector bundle of rank $r$ on $X$. Assume $d \ge 2$. Then we have $$(r-1)(c_1(E)^2 \cdot H^{d-2}) \le 2r(c_2(E)\cdot H^{d-2})$$ if (1) $r < 3$ or (2) $d = 2$.

My question is, are there similar, further results that relate numerical invariants of a nonsingular projective variety in positive characteristic? In particular, it would help me a great deal if there was one involving also the characteristic $p=\mathrm{char}(k)$ itself.

It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can find the following

Theorem 1. Assume that $p = \mathrm{char}(k) > 0$. Let $X$ be a $d$-dimensional nonsingular projective $k$-variety with an ample line bundle $H$. Let $E$ be a $p$-semistable (with respect to $H$) vector bundle of rank $r$ on $X$. Assume $d \ge 2$. Then we have $$(r-1)(c_1(E)^2 \cdot H^{d-2}) \le 2r(c_2(E)\cdot H^{d-2})$$ if (1) $r < 3$ or (2) $d = 2$.

My question is, are there similar, further results that relate numerical invariants of a nonsingular projective variety in positive characteristic? In particular, it would help me a great deal if there was one involving also the characteristic $p=\mathrm{char}(k)$ itself.