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

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