ok, this is trivial -- easy application of maximum principle
sorry for bothering
basically I need to justify in this very particular setup
let be analytic functions on some fixed annulus where . It is given that tend to the constant function 1 uniformly on compact sets of and of (so everywhere except of the unit circle). Prove that tend to 1 for on the unit circle as well.
WLOG take . we can choose , , and so is what we need
it's clearly crucial that are analytic and converge both inside and outside the unit circle (as the counterexample suggests), but I can't find a way to use it