On fully operator Lipschitz functions

Let A(D) be the disc algebra of all continuous complex-valued functions on the unit disc D holomorphic in its interior. Functions from A(D) act on the set of all contraction operators (‖A‖⩽1) on Hilbert spaces. It is proved that the following classes of functions from A(D) coincide: (1) the class of operator Lipschitz functions on the unit circle T; (2) the class of operator Lipschitz functions on D; and (3) the class of operator Lipschitz functions on all contraction operators. A similar result is obtained for the class of operator C2-Lipschitz functions from A(D).