2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62291If $X$ 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.)25 pages, the final version, to appear in "Journal of the Mathematical Society of Japan"Algebraic Geometry14E05text