Schur multiplier norms for Loewner matrices

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 $\mathrm{\mid }\mathrm{\mid }[A,f(B)]\mathrm{\mid }{\mathrm{\mid }}_q\mathrm{/}\mathrm{\mid }\mathrm{\mid }[A,B]\mathrm{\mid }{\mathrm{\mid }}_q$. We also consider operator monotone functions, for which sharper bounds are obtained.