We know that if an $n$-dimensional Fano manifold admits a Kahler-Einstein metric, it satisfies the following Chern number inequality $$nc_1^n\leq 2(n+1)c_2c_1^{n-2}.$$

My question is whether there exists a Fano manifold which does not satisfy this inequality (of course in this case this Fano manifold can not have any K-E metric). I calculated some examples which have been famously known that they don't support K-E metrics and find out that all of them still satisfy this inequality.

Thanks in advance.