On the geography of Gorenstein minimal 3-folds of general type

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $\chi ({\omega }_X)$ and $p_g(X)$:

1. Assume that the canonical map ${\mathrm{\Phi }}_{\mathrm{\mid }{K}_X\mathrm{\mid }}$ is of fiber type. Let $F$ be a smooth model of a generic irreducible component in the general fiber of ${\mathrm{\Phi }}_{\mathrm{\mid }{K}_X\mathrm{\mid }}$. Then the birational invariants of $F$ are bounded from above.
2. If $X$ is nonsingular, then $c_1^3\le 1\mathrm{/}27c_1{c}_2+10\mathrm{/}3$ where $c_1$, $c_2$ are Chern invariants of $X$.