The inverse mean problem of geometric mean 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])A=X+Y-2(X-1+Y-1)-1,B=X#Y,by proving its equivalence to the well-known nonlinear matrix equation X = T − BX−1B where T=12(A+A#(A+8BA-1B)) is the unique positive definite solution of X = A + 2BX−1B. The inverse mean problem is solvable if and only if B ⩽ A.