Improved inequalities for balanced incomplete block designs

It is known that there are some lower bounds for the number of blocks in a balanced incomplete block design (BIBD). Especially, Fisher's inequality b?v is well-known for a BIBD with parameters v, b, r, k and λ. Fisher's inequality can be improved upon if one puts additional restrictions on a BIBD. Artificial restrictions are infinite in number so is the number of new bounds. The condition of non-symmetry on the design discussed here is a very simple restriction. The main purpose of this paper is to give improvements of inequalities for BIBDs with the only condition of non-symmetry. Improved inequalities appear to be the best for any non-symmetrical BIBD.