It is known that the function (Formula presented.) is an operator Lipschitz function on the real line (Formula presented.). 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 (Formula presented.) is also operator Lipschitz if f(0) = 0. The function f can be defined on an arbitrary closed subset of the complex plane (Formula presented.). Moreover, the linear fractional transformation (Formula presented.) can be replaced by every linear fractional transformation φ{symbol}. In this case, we assert that the function (Formula presented.) is operator Lipschitz for every operator Lipschitz function f provided that f(φ{symbol}(∞)) = 0. Bibliography: 12 titles.