Operator Lipschitz functions and linear fractional transformations

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 $f(\varphi(\infty))=0$.