Frechet derivatives for operator monotone functions

Let phi be an operator monotone function from the interval (0, infinity) into itself whose analytic continuation is univalent on the union D of the interval and upper and lower half-planes, and psi its inverse. For a square matrix A whose all eigenvalues are in phi(D), the solution X of the equation D psi(A)(X) = Y, X = D phi(psi(A))(Y), is given explicitly; here D phi(A) is the Frechet derivative of phi at A. An example related to Dyson's expansion is also discussed. (C) 2013 Elsevier Inc. All rights reserved.