Geodesic
Operator Lipschitz functions in several variables and M\"obius transformations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 5-32

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It is proved that if $f$ is an operator Lipschitz function defined on $\mathbb R^n$, then the function $\dfrac{f\circ\varphi}{\|\varphi'\|}$ is also operator Lipschitz for every Möbius transformations $\varphi$ with $f(\varphi(\infty))=0$. Here $\|\varphi'\|$ denotes the operator norm of the Jacobian matrix $\varphi'$. Similar statements are obtained also for operator Lipschitz functions defined on closed subsets of $\mathbb R^n$. 
@article{ZNSL_2014_424_a0,
     author = {A. B. Aleksandrov},
     title = {Operator {Lipschitz} functions in several variables and {M\"obius} transformations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--32},
     publisher = {mathdoc},
     volume = {424},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a0/}
}
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A. B. Aleksandrov. Operator Lipschitz functions in several variables and M\"obius transformations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 5-32. http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a0/