A generalized Schwarz lemma at the boundary (Q2731920)

\textit{D. Burns} and \textit{S. Krantz} [J. Am. Math. Soc. 7, No. 3, 661-676 (1994; Zbl 0807.32008)] proved, in particular, the following beautiful boundary version of Schwarz's lemma. Let \(\varphi\) be an analytic function mapping the unit disk \(\mathbb{D}\) into itself such that \(\varphi(z)=z+O((z-1)^4)\) as \(z\to 1\). Then \(\varphi(z)=z\) on the unit disk \(\mathbb{D}\). In the paper under review, the author's nice generalization of the above lemma extends to finite Blaschke products.