Geodesic
Operator Lipschitz functions and linear fractional transformations
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 5-52
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It is known that the function $t^2\sin\frac1t$ is an operator Lipschitz function on the real line $\mathbb R$. We prove that the function $\sin$ can be replaced by any operator Lipschitz function $f$ with $f(0)=0$. In other words, for every operator Lipschitz function $f$ the function $t^2 f(\frac1t)$ is also operator Lipschitz if $f(0)=0$. The function $f$ can be defined on an arbitrary closed subset of the complex plane $\mathbb C$. Moreover, the linear fractional transformation $\frac1t$ can be replaced by every linear fractional transformation $\varphi$. In this case, we assert that the function $\dfrac{f\circ\varphi}{\varphi'}$ is operator Lipschitz for every operator Lipschitz function $f$ provided $f(\varphi(\infty))=0$. 
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     author = {A. B. Aleksandrov},
     title = {Operator {Lipschitz} functions and linear fractional transformations},
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A. B. Aleksandrov. Operator Lipschitz functions and linear fractional transformations. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 40, Tome 401 (2012), pp. 5-52. http://geodesic.mathdoc.fr/item/ZNSL_2012_401_a0/

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