An estimate of quasi-arithmetic mean for convex functions

For a selfadjoint operator A on a Hilbert space H and a normalized positive linear map Phi, a quasi-arithmetic mean is defined by phi^-1(Phi(phi(A)) for a strictly monotone function phi. In this paper, we shall show an order relation among quasi-arithmetic means for convex functions through positive linear maps and its complementary problems, in which we use the Mond-Pečarić method for convex functions.