Trivial Estimate for Cyclotomic Polynomials

Theorem

Let $n \ge 1$ be a natural number.

Let $\Phi_n$ be the $n$th cyclotomic polynomial.

Let $\phi$ be the Euler totient function.

Let $z \in \C$ be a complex number.

Then:

$\size {\size z - 1}^{\map \phi n} \le \size {\map {\Phi_n} z} \le \paren {\size z + 1}^{\map \phi n}$

where:

the first inequality becomes an equality only if:

$n = 1$ and $z \in \R_{\ge 0}$

or:

$n = 2$ and $z \in \R_{\le 0}$

the second inequality becomes an equality only if:

$n = 1$ and $z \in \R_{\le 0}$

or:

$n = 2$ and $z \in \R_{\ge 0}$

Proof

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