# What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

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?

• I think it's just a sufficient condition. There are probably many polynomials which do not satisfy $|a_N| \geq \frac{1}{2} \sum |a_n|$ but which have zeros on the unit circle. – Antonio Vargas Jul 13 '14 at 16:58
• @AntonioVargas I think you are correct. The condition mentioned above is merely a sufficient condition. – sauravrt Jul 14 '14 at 22:34