Putnam 2018, Problem B-2

Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C} : |z| \leq 1\}$.

Solution:
First solution. Note first that $f_n(1)>0$, so 1 is not a root of $f_n$. Next, note that
$$(z-1)f_n(z)=z^n+\cdots+z-n;$$
however, for $|z|\leq 1$, we have $|z^n+\cdots+z|\leq n$ by the triangle inequality; equality can only occur if $z,\ldots,z^n$ have 1 and the same argument, which only happens for $z=1$. Thus there can be no root of $f_n$ with $|z|\leq 1$.

Second solution. (by Karl Mahlburg) Define the polynomial
$$g_n(z)=nz^{n-1}+\cdots+2z+1$$
and note that $z^{n-1}g_n(z^{-1})=f_n(z)$. Since $f_n(0)\neq 0$, to prove the claim it is equivalent to show that $g_n$ has no roots in the region $|z|\geq 1$.

Now note that $g_n(z)=h_n'(z)$ for
$$h_n(z)=z^n+\cdots+z+1,$$
a polynomial with roots $e^{2\pi ij/(n+1)}$ for $j=0,\ldots,n$. By the Gauss-Lucas theorem, the roots of $g_n$ lie in the convex hull of the roots of $h_n$, and moreover cannot be vertices of the convex hull because $h_n$ has no repeated roots. This implies the claim.

Remark. Yet another approach is to use the Eneström-Kakeya theorem: if $P_n(z)=a_0+\cdots+a_nz^n$ is a polynomial with real coefficients satisfying $|a_n|\geq\cdots\geq|a_0|>0$, then the roots of $P_n(z)$ all satisfy $|z|\leq 1$. Namely, applying this to the polynomial $g_n(z/c)$ for $c=n/(n-1)$ shows that the roots of $g_n$ all satisfy $|z|\leq 1/c$.

Remark. For a related problem, see problem A5 from the 2014 Putnam competition.