Bounds for the dimension of Lie algebras

In 1993, moneyhun showed that if l is a lie algebra such that dim(l/z(l)) = n, then dim(l^2) ≤ 1/2n(n-1). the author and saeedi investigated the converse of moneyhun's result under some con-ditions. in this paper, we extend their results to obtain several upper bounds for the dimension of a lie algebra l in terms of dimension of l^2, where l^2 is the derived subalgebra. moreover, we give an upper bound for the dimension of the c-nilpotent multiplier of a pair of lie algebras.