Trivial Estimate for Cyclotomic Polynomials

Theorem
Let $n\geq1$ 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 $||z|-1|^{\phi(n)} \leq \left|\Phi_n(z)\right| \leq (|z|+1)^{\phi(n)}$, where:
 * The first inequality becomes an equality only if $n=1$ and $z\in\R_{\geq0}$ or $n=2$ and $z\in\R_{\leq0}$
 * The second inequality becomes an equality only if $n=1$ and $z\in\R_{\leq0}$ or $n=2$ and $z\in\R_{\geq0}$