Suppose that $p(z)$ is a polynomial of degree $k$ with leading coefficient equal to $1$ , moreover assume that all the zeros of p lie in the unit disc, then for z suffieciently large
$$|p(z)| \ge \frac{9|z|^{k}}{10}$$
The poof that was given was ; The result follows immediately from
$$|1/z| \lt \frac{1}{(10k)(max|a_{j}|+1)}$$
But I don't understand where they came up with this from and how it implies the result? Can anyone help me to understand? I am really not sure, I tried to think about maybe radius of convergence.
say we write
$$p(z)=a_{0}+a_{1}z+...+a_{k-1}z^{k-1}+z^{k}$$
$$p(z)=z^{k}(1+a_{k-1}/z+...+a_{0}/z^{k})$$
I thought maybe since we are told that zeros are only in the unit circle, then for z large ie z away from the unit circle and $z=0$ then p(z) is maybe holomorphic and so has a convergent power series?
But I am really not sure and very confused.
Answer
Can you show that for $z$ sufficiently large the inequality below holds?
$$\left(1+\frac{a_{k-1}}{z}+...+\frac{a_{0}}{z^{k}}\right)\geq \frac{9}{10}$$
What happens as $z \to \infty$?
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