Stolz angle limit of a certain class of self-mappings of the unit disk (Q435182)

Let \(f\) be a mapping of the open unit disk \(\mathbf U\) onto itself having a non-singular differentiable extension to the boundary point 1 which is a fixed point of \(f\). For \(a \in \mathbf U\) let \(p\) and \(q\) be Möbius transformations of the unit disk onto itself such that \(p(0)=a\) and \(q(f(a))=0\). It is proved that the Stolz angle limit of \(p\circ f\circ q\) when \(a\to 1\) is a diffeomorphic self-mapping \(g\) of the unit disk which is a conjugate of an affine transformation. The convergence is almost uniform in \(\mathbf U\).