Fréchet derivatives for operator monotone functions

Let φ be an operator monotone function from the interval (0,∞) into itself whose analytic continuation is univalent on the union D of the interval and upper and lower half-planes, and ψ its inverse. For a square matrix A whose all eigenvalues are in φ(D), the solution X of the equation Dψ(A)(X)=Y, X=Dφ(ψ(A))(Y), is given explicitly; here Dφ(A) is the Fréchet derivative of φ at A. An example related to Dyson's expansion is also discussed.