Bounds for the zeros of Collatz polynomials, with necessary and sufficient strictness conditions

In a previous work, we introduced the Collatz polynomials; these are the polynomials $left[P_N(z)right]_{Ninmathbb{N}}$ such that $left[z^0right]P_N = N$ and $left[z^{k+1}right]P_N = cleft(left[z^kright]P_Nright)$, where $c:mathbb{N}rightarrow mathbb{N}$ is the Collatz function $1rightarrow 0$, $2nrightarrow n$, $2n+1rightarrow 3n+2$ (for example, $P_5(z) = 5 + 8z + 4z^2 + 2z^3 + z^4$). In this article, we prove that all zeros of $P_N$ (which we call Collatz zeros) lie in an annulus centered at the origin, with outer radius 2 and inner radius a function of the largest odd iterate of $N$. Moreover, using an extension of the Enestrom-Kakeya Theorem, we prove that $|z| = 2$ for a root of $P_N$ if and only if the Collatz trajectory of $N$ has a certain form; as a corollary, the set of $N$ for which our upper bound is an equality is sparse in $mathbb{N}$. Inspired by these results, we close with some questions for further study.