Apologies in advance if this turns out to be simple.  So far I haven't found a proof or a reference.

Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, using what should be clear notations for the $n$th cyclotomic polynomial and Euler's totient function: Given $p \gt 1$, is

$$ \mid \Phi_n(p)/p^{\phi(n)} \mid \lt p/(p-1)$$
for every $n$?

Indeed, if $n$ is a power of a prime $q$, we have the left hand quantity bounded by $p^{n/q}/(p^{n/q} -1)$, and the lim sup over all primes $n$ achieves $p/(p-1)$. The case for composite $n$ is not clear to me, thus the question, but I would hope for a tighter bound (perhaps involving the smallest prime power factor of $n$) than $p/(p-1)$.  

An equivalent question asks to verify the bound
on $\Phi_n(1/p)$.  Of course the product of such quantities (to an appropriate power as $n$ runs over divisors of some $m$) will satisfy the bound, but this does not seem to help. If there is a reference offered that says (something like) the coefficients of cyclotomic polynomials grow slowly enough to exhibit the bound, I will read that.  I am hoping for a simpler proof than that.

I am looking at (the moral equivalent of)  prime factors of $\Phi_n(p)$ and wanted to make sure these values aren't much bigger than I think they are.
I would be satisfied with a coarse bound (replace $p/(p-1)$ by $2$, say), but
I think much more can be said.

Gerhard "Wants To Stop Spinning Head" Paseman, 2015.10.19