The Hadamard product for the weighted Karcher means

The weighted Karcher mean of positive definite matrices is defined as the unique minimizer of the weighted sum of squares of the Riemannian distances to each of given points. Using the well-known connection between the tensor product and the Hadamard product, we show that the Hadamard product of weighted Karcher means for permuted tuples with fixed weight is bounded by the Hadamard product of given positive definite matrices. It generalizes the results for the case of two-variable geometric means and Sagae–Tanabe inductive means.