Reviewing bounds on the circuit size of the hardest functions

In this paper we review the known bounds for L(n), the circuit size complexity of the hardest Boolean function on n input bits. The best known bounds appear to be 2nn(1+lognn−O(1n))⩽L(n)⩽2nn(1+3lognn+O(1n)). However, the bounds do not seem to be explicitly stated in the literature. We give a simple direct elementary proof of the lower bound valid for the full binary basis, and we give an explicit proof of the upper bound valid for the basis {¬,∧,∨}.