Functions on noncommuting self-adjoint operators under perturbation

I am going to define functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$.Then I am going to study Lipschitz type estimates of the form $$\|f(A_1,B_1)-f(A_2,B_2)\|\le\operatorname{const}\max\{\|A_1-A_2\|,\|B_1-B_2\|\}$$ for different norms. It turns out that such Lipschitz type estimates hold in the Schatten--von Neumann norms $S_p$ with $1\le p\le2$ for bounded functions $f$ on the plane whose Fourier transform is compactly supported. On the other hand, such Lipschitz type estimates do not hold in the Schatten--von Neumann norms $S_p$ with $p>2$, nor in the operator norm.