Inequalities of Noether type for 3-folds of general type

If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge 2\mathrm{/}3(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. (A noether type of inequality lies, by all means, on the other side of the Miyaoka-Yau inequality from the geographical point of view.)