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 L2, 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.