On the commutator modulus of continuity for operator monotone functions

Let $f \geq 0$ be operator monotone on $[0, \infty)$. In this paper we prove that for any unitarily-invariant norm $|||-|||$ on $M_n(\mathbb{C})$ and matrices $A, B, X \in M_n(\mathbb{C})$ with $A, B \geq 0$ and $|||X||| \leq 1$, $$|||f(A)X-Xf(B)||| \leq C f(|||AX-XB|||)$$ for $C < 1.01975$. We do this by reducing this inequality to a function approximation problem and we choose approximate minimizers. This is much progress toward the conjecturally optimal value of $C=1$ which is known only in the case of the Hilbert-Schmidt norm. When $|||-|||$ is the the operator norm $||-||$, we obtain a great reduction of the previously known estimate of $C = 1.25$. We further prove that for $|||X||| \leq 1$, $$|||A^{1/2}X-XB^{1/2}||| \leq 1.00891 |||AX-XB|||.$$ This is a great improvement toward the conjecture of G. Pedersen that this inequality for $|||-|||$ being the operator norm holds with $C = 1$. We discuss other related inequalities, including some sharp commutator inequalities. We also prove a sharp equivalence inequality between the operator modulus of continuity and the commutator modulus of continuity for continuous functions on $\mathbb{R}$.