For a resolvable balanced incomplete block design (BIBD), Bose (1942) obtained an inequality $b \geqq v + r - 1$. Stanton (1957) showed that this inequality is equivalent to an inequality $r \geqq \lambda + k$. The main purpose of this note is to improve Bose's inequality to $b \geqq 2\nu + r - 2$ for a resolvable BIBD which is not affine resolvable.
"An Improved Inequality for Balanced Incomplete Block Designs." Ann. Math. Statist. 42 (4) 1448 - 1449, August, 1971. https://doi.org/10.1214/aoms/1177693259