A weighted version of the geometric mean of $k (geq 3)$ positive invertible operators is given. For operators $A_1, ..., A_k$ and for nonnegative numbers $a_1, ..., a_k$ such that $sum_{i=1}^k a_i = 1,$ we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to $A^{a_1}_1 cdots A^{a_k}_k$ if $A_1, ..., A_k$ commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

Keywords: positive operator, weighted geometric mean, arithmetic-geometric
mean inequality, reverse inequality