A construction of an annular function with prescribed local behavior.
A condition of Schmets and Valdivia for a boundary point of a domain in the complex plane to be regularly asymptotic is ameliorated.
We improve a theorem of C. L. Belna (1972) which concerns boundary behaviour of complex-valued functions in the open upper half-plane and gives a partial answer to the (still open) three-segment problem.
In this note we present a simple proof of a theorem of Hornblower which characterizes those functions analytic in the open unit disk having asymptotic values at a dense set in the boundary. Our method is based on a kind of ∂-mollification and may be of use in other problems as well.