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

Abstract

Let f 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 f. For a∈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∘f∘q when a→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 U.