Problem 8

Exercise 4.6 (Polynomial ideals TODO)  

Proof. The estimate

$$\vert f(z)\vert \leq A(1+\vert z\vert^{-s})$$ (4.29)

implies that

$$\vert z^sf(z)\vert \leq A\vert z\vert^s + A,$$ (4.30)

which indicates $z^sf(z)$ is a polynomial, say:

$$z^sf(z) = a_0 + \cdots + a_rz^r$$ (4.31)

so that

$$f(z) = \frac{a_0}{z^s} + \cdots + a_rz^{r-s}$$ (4.32)

Conversely, if $f(z)$ is a sum as written above, then take $A=\sum\vert a_k\vert$. By the triangle inequality

$$\vert f(z)\vert \leq \vert a_0\vert\vert z\vert^{-s} + \vert a_1\vert\vert z^{1-s}\vert + \cdots + \vert a_r\vert\vert z^{r-s}\vert$$ (4.33)

$\qedsymbol$