  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 $\Phi_{|K_X|}$ is of fiber type. Let $F$ be
a smooth model of a generic irreducible component in the general fiber of
$\Phi_{|K_X|}$. Then the birational invariants of $F$ are bounded from above.
  2) If $X$ is nonsingular, then $c_1^3\leq {1/27} c_1c_2+{10/3}$ where $c_1$,
$c_2$ are Chern invariants of $X$.
