  In this paper, we prove a stronger form of the Bogomolov-Gieseker (BG)
inequality for stable sheaves on two classes of Calabi-Yau threefolds, namely,
weighted hypersurfaces inside the weighted projective spaces $\mathbb{P}(1, 1,
1, 1, 2)$ and $\mathbb{P}(1, 1, 1, 1, 4)$. Using the stronger BG inequality as
a main technical tool, we construct open subsets in the spaces of Bridgeland
stability conditions on these Calabi-Yau threefolds.
