This should help, but it's late here:let $d = \phi(n)$ and assume $n>2$.Then the primitive $n$-th roots of unity occur in complex conjugate pairs, and the GM-AM inequality gives $\Phi_n(p)/p^d \leq (1 + \frac{1}{p^2}+ \frac{2}{dp})^{d/2}$, since the sum of the primitive $n$-th roots of unity is at most $1$. This is less than 
$e^{1/p}(1+ \frac{1}{p^{2}})^{d/2}$.