Characterization of finite shift via Herglotz's representation

A complete characterization of parabolic self-maps of finite shift is given in terms of their Herglotz's representation. This improves a previous result due to Contreras, Díaz-Madrigal, and Pommerenke. We also derive some consequences for the rate of convergence of these functions to their Denjoy-Wolff point, improving a related result of Kourou, Theodosiadis, and Zarvalis for the continuous setting.