Some properties of self-inversive polynomials

Abstract:

A complex polynomial is called self-inversive [5, p. 201] if its set of zeros (listing multiple zeros as many times as their multiplicity indicates) is symmetric with respect to the unit circle. We prove that if $P$ is self-inversive and of degree $n$ then $||P’|| = \tfrac {1}{2}n||P||$ where $||P’||$ and $||P||$ denote the maximum modulus of $P’$ and $P$, respectively, on the unit circle. This extends a theorem of P. Lax [4]. We also show that if $P(z) = \Sigma _{j = 0}^n{a_j}{z^j}$ has all its zeros on $|z| = 1$ then $2\Sigma _{j = 0}^n|{a_j}{|^2} \leqq ||P|{|^2}$. Finally, as a consequence of this inequality, we show that when $P$ has all its zeros on $|z| = 1$ then ${2^{1/2}}|{a_{n/2}}| \leqq ||P||$ and $2|{a_j}| \leqq ||P||$ for $j \ne n/2$. This answers in part a question presented in [3, p. 24].