The inverse mean problem of geometric and contraharmonic means

In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed in [W.N. Anderson, M.E. Mays, T.D. Morley, G.E. Trapp, The contraharmonic mean of HSD matrices, SIAM J. Algebra Disc. Meth. 8 (1987) 674-682]) [GRAPHICS] by proving its equivalence to the well-known nonlinear matrix equation X = T - BX-1B where T = 1/2(A + A#(A + 8BA(-1) B)) is the unique positive definite solution of X = A + 2BX(-1) B. The inverse mean problem is solvable if and only if B <= A. (c) 2005 Elsevier Inc. All rights reserved.