  We study upper bounds on the Schur multiplier norm of Loewner matrices for
concave and convex functions. These bounds then immediately lead to upper
bounds on the ratio of Schatten $q$-norms of commutators
$||\;[A,f(B)]\;||_q/||\;[A,B]\;||_q$. We also consider operator monotone
functions, for which sharper bounds are obtained.
