$z^{-m}$ is not a uniform limit of polynomials on an annulus

1. Let \(p ( z )\) be a polynomial, \(R > 0\) any positive number, and \(m \geq 1\) an integer. Let \(M _ { R } = \operatorname* { s u p } \{ | z ^ { m } p ( z ) - 1 | : | z | = R \}\) . Show that \(M _ { R } > 1\) •

2. Let \(m \geq 1\) be an integer and \(K = \{ z \in \mathbb { C } : r \leq | z | \leq R \}\) where \(r < R .\) . Show (i) using (1) as well as, (ii) without using (1) that there exists a positive number \(\varepsilon _ { 0 } > 0\) such that for each polynomial \(p ( z )\) ,

\[ \operatorname* { s u p } \{ | p ( z ) - z ^ { - m } | : z \in K \} \geq \varepsilon _ { 0 } . \]