If I have two complex functions defined by power series $A(z) = \sum a_n z^n $, $B(z) = \sum b_n z^n$ with $|a_n| \ge |b_n|$ for all $n$, and I know that $A$ converges in some set $U_1$ and defines a function of $z$ with an analytic extension to set $U_2$, then of course $B$ converges in $U_1$ also, but does this imply that $B$ has an analytic extension to $U_2$?
It would be nice if this were true, but I am a bit concerned because it's not hard to imagine a situation in which $B(z)$ that slightly shifts the region where $A(z)$ fails to have an analytic continuation. Can anyone help me think of a concrete counter example, or give an idea towards a proof?