A reverse inequality for the weighted geometric mean due to Lawson–Lim

In this note, we present an alternative proof of the power convergence of the symmetrization procedure on the weighted geometric mean due to Lawson and Lim in [J. Lawson and Y. Lim, A general framework for extending means to higher orders, preprint] by using a limiting process due to Ando-Li-Mathias in [T. Ando, C.-K. Li, R. Mathias, Geometric means, Linear Algebra Appl. 385 (2004) 305–334]. As applications, we obtain a reverse of the weighted arithmetic-geometric mean inequality of n-operators via Kantorovich constant: For any positive integer n⩾2, let A1,A2,…,An be positive invertible operators on a Hilbert space H such that 0