Proposition 2.1

  Suppose that is a power series of the form

Then for any r, 0 < r < 1, has

 

zeros in .

Proof

We apply Jensen's theorem (Theorem 3.61 of [17]). If are the zeros in |z| < R, where r < R < 1, then we find that

 

since . Therefore, if m is the number of zeros in |z| < r, we have

 

Since

we obtain

We now choose , and this yields the bound . (Better bounds can be obtained by selecting R more carefully or estimating the integral of in Jensen's theorem better.)