The class of holomorphic self-mappings of the open unit disk (which are contractions with respect to the Poincaré metric) admits a natural extension to the class of holomorphic pseudo-contractions. In this paper, we study various inequalities involving the values of derivatives of holomorphic pseudo-contractions at fixed points (particularly, at the Denjoy–Wolff fixed point).
Alpay, D., Bolotnikov, V. & Shoikhet, D. Pseudo-contractions, rigidity, fixed points and related questions. Anal.Math.Phys. 10, 7 (2020) https://doi.org/10.1007/s13324-019-00346-2