We obtain higher derivative generalizations of the Schwarz-Pick inequality for analytic self-maps of the unit disk as a consequence of recent characterizations of boundedness and compactness of weighted composition operators between Bloch-type spaces.
MacCluer, Barbara D.; Stroethoff, Karel; and Zhao, Ruhan, "Generalized Schwarz-Pick Estimates" (2002). Mathematics Faculty Publications. 4.
B. MacCluer, K. Stroethoff and R. Zhao: Generalized Schwarz-Pick estimates, Proc. Amer. Math. Soc., 131(2003), 593-599.
First published in Proceedings of the American Mathematical Society in 2003, published by the American Mathematical Society