Asked by Ho Pak.

Suppose and are holomorphic in a region containing the disc . Suppose that has a simple zero at and vanishes nowhere else in . Let .

Show that if is sufficiently small, then

(a) has a unique zero in , and

(b) if is this zero, the mapping is continuous.

Would like to know how to do (b).

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(I can’t post the comment properly so I tried to break it into two. )

(a) on the circle if is small enough, so by Rouche’s theorem has a unique zero in the unit disk.

(b) I only prove that it is continuous at , the general case is the same (by a “translation” argument) but only notationally more difficult.

Let’s denote , clearly . Now, suppose there exists such that for all , there is a with . We can assume that (by taking a subsequence) with . Then as , by letting , we have , a contradiction.