  We prove a Bogomolov-Gieseker type inequality for the third Chern characters
of stable sheaves on Calabi-Yau 3-folds and a large class of Fano 3-folds with
given rank and first and second Chern classes. The proof uses the spreading-out
technique, vanishings from the tilt-stability conditions, and Langer's
estimation theorem of the global sections of torsion free sheaves. In
particular, the result implies that the conjectural sufficient conditions on
the Chern numbers for the existence of stable sheaves on a Calabi-Yau 3-fold by
Douglas-Reinbacher-Yau needs to be modified.
