  If $X$ is a smooth complex projective 3-fold with ample canonical divisor
$K$, then the inequality $K^3\ge {2/3}(2p_g-7)$ 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.)
