Analysis: fixed point, contraction mapping

[complexnumber]

Let p,q : \mathbb{C} \to \mathbb{C} be
defined by
<br /> \begin{align*}<br /> p(z) =&amp; z^7 + z^3 - 9z - i, \\<br /> q(z) =&amp; \frac{z^7 + z^3 - i}{9}<br /> \end{align*}<br />

1. Prove that p has a zero at z_0 if and only if z_0 is a
fixed point for q.

If z_0 is a fixed point for q then
<br /> \begin{align*}<br /> q(z_0) = \frac{z_0^7 + z_0^3 - i}{9} =&amp; z_0 \\<br /> z_0^7 + z_0^3 - i =&amp; 9 z_0 \\<br /> z_0^7 + z_0^3 - 9 z_0 - i =&amp; p(z_0) = 0<br /> \end{align*}<br />
Hence z_0 is a zero for p.

If p has a zero at z_0 then
<br /> \begin{align*}<br /> p(z_0) = z_0^7 + z_0^3 - 9 z_0 - i =&amp; 0 \\<br /> z_0^7 + z_0^3 - i =&amp; 9 z_0 \\<br /> \frac{z_0^7 + z_0^3 - i}{9} =&amp; q(z_0) = z_0<br /> \end{align*}<br />
Hence z_0 is a fixed point for q.

2. Hence or otherwise show that p has exactly one zero in
the closed unit disk D = \{ z \in \mathbb{C} : |z| \leq 1 \}.

How can I solve this?

3. Where are the other zeros?

Does this follow part 2?

 

[Tinyboss]

Show that q is a contraction map from D into D, i.e. that |q(w)-q(z)|&lt;\alpha|w-z| with \alpha&lt;1. Then use a theorem about contraction maps.

 

[complexnumber]

Show that q is a contraction map from D into D, i.e. that |q(w)-q(z)|&lt;\alpha|w-z| with \alpha&lt;1. Then use a theorem about contraction maps.

What metric should I use for complex number space? The question did not mention any metric function at all.

 

[complexnumber]

In D the derivative q&#039; is not less than 1.

<br /> \begin{align*}<br /> |q&#039;(z)| = <br /> |\frac{7z^6 + 3z^2}{9}| \leq \frac{1}{9}<br /> (|7z^6| + |3z^2|) \leq \frac{1}{9} (7 + 3) = \frac{10}{9}.<br /> \end{align*}<br />

Is q : D \to D is a contraction mapping in D?