Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition for all roots of $P(z)$ to be on the unit circle is that all zeros of $P'(z)$ lie on $|z|<1$.
Also they showed that if
$|a_N| \geq \frac{1}{2}\sum\limits_{n=1}^{N-1}|a_n|$,
$P(z)$ has all roots on unit circle.
I'm unable to see how this condition holds when all coefficients $a_n = 1$ or $|a_n| = 1$ i.e. uni-modular coefficients and conjugate reciprocal.
Please help me understand what I'm missing here?