Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov
[1, Theorem 5] proved that c21 [les ] 4c2.
This was improved to c21 [les ] 3c2 by
Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality
c21 [les ] 3c2 is attained,
for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only
possibility). Further generalizations of inequalities for Chern classes for some singular
surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6],
Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12],
Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].
In  we introduced Chern classes of reflexive sheaves, using Wahl's local Chern
classes of vector bundles on resolutions of surface singularities. Here we apply them
to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.