A neighborhood of the unit circle

Lemma 3.1

  If is compact, , |z| < 1, and

 

then every element of B is expressible in the form

 

In particular, if , then .

Proposition 3.1

  If , , then .

[Proof]

Lemma 3.2

  If is compact, , , and

 

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

[Proof]

Lemma 3.3

  If for some , , |x| =1 and

 

then

[Proof]

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 .

[Proof]

Lemma 3.5

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

[Proof]

Proposition 3.3

  .

[Proof]

We now combine all the results of this section.

Theorem 3.1

  There is an open neighborhood of contained in .

[Proof]

Corollary 3.1

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

[Proof]