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 P is self-inversive and of degree n n then | | P ′ | | = 1 2 n | | P | | ||P’|| = \tfrac {1}{2}n||P|| where | | P ′ | | ||P’|| and | | P | | ||P|| denote the maximum modulus of P ′ P’ and P P , respectively, on the unit circle. This extends a theorem of P. Lax [4]. We also show that if P ( z ) = Σ j = 0 n a j z j P(z) = \Sigma _{j = 0}^n{a_j}{z^j} has all its zeros on | z | = 1 |z| = 1 then 2 Σ j = 0 n | a j | 2 ≦ | | P | | 2 2\Sigma _{j = 0}^n|{a_j}{|^2} \leqq ||P|{|^2} . Finally, as a consequence of this inequality, we show that when P P has all its zeros on | z | = 1 |z| = 1 then 2 1 / 2 | a n / 2 | ≦ | | P | | {2^{1/2}}|{a_{n/2}}| \leqq ||P|| and 2 | a j | ≦ | | P | | 2|{a_j}| \leqq ||P|| for j ≠ n / 2 j \ne n/2 . This answers in part a question presented in [3, p. 24].