For a selfadjoint operator $A$ on a Hilbert space $H$ and a normalized positive linear map $\Phi$, a quasi-arithmetic mean is defined by $\varphi^{-1} \left( \Phi (\varphi(A)) \right)$ for a strictly monotone function $\varphi$. 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-Pe\v{c}ari\'{c} method for convex functions.