  Bose proved the inequality $b\geq v+r-1$ for resolvable balanced incomplete
block designs (RBIBDs) and Kageyama improved it for RBIBDs which are not affine
resolvable. In this note we prove a new lower bound on the number of blocks $b$
that holds for all BIBDs. We further prove that for a significantly large
number of BIBDs our bound is tighter than the bounds given by the inequalities
of Bose and Kageyama.
