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# A neighborhood of the unit circle

In this section we prove that an open neighborhood of is contained in .

### Lemma 3.1

If is compact, , |z| < 1, and

then every element of B is expressible in the form

In particular, if , then .

If , , then .

### Lemma 3.2

If is compact, , , and

where int denotes the interior of S, then there is a neighborhood N of x such that

### Lemma 3.3

If for some , , |x| =1 and

then

### Proposition 3.2

If |x| =1, , then .

### [Proof]

Proving is trickier, because it will not do to take B as a disc of radius if is small compared to . We will instead take B as a parallelogram that becomes flatter and flatter as . The following two lemmas will be used in verifying the condition of Lemma 3.1.

### Lemma 3.4

Let . Let . Then for

contains .

### Lemma 3.5

Let T, be as in Lemma 3.4. Let B be the square with vertices . Then for ,

.

### [Proof]

We now combine all the results of this section.

### Theorem 3.1

There is an open neighborhood of contained in .

### Corollary 3.1

If for sufficiently small then z is a multiple zero of some 0,1 power series.

### [Proof]

Contents Next: is connected Up: No Title Previous: Bounds and locations