IMC 2021 · Day 2 · P7

Hint: Apply the maximum principle or the Cauchy formula to a suitable function $f(z) q(z)$.

Let $q(z) = z^n \cdot p(1/z)$, or more explicitly, if $$p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0,$$ let $$q(z) = 1 + a_{n-1} z + \dots + a_0 z^n.$$ Note that for $|z| = 1$ we have $1/z = \bar{z}$ and hence $|q(z)| = |p(z)|$. Hence by the maximum principle or the Cauchy formula for the product of $f$ and $q$, it follows that $$\bigl| f(0) \bigr| = \bigl| f(0) q(0) \bigr| \le \max_{|z|=1} \bigl| f(z) q(z) \bigr| = \max_{|z|=1} \bigl| f(z) p(z) \bigr|.$$