## Some properties of self-inversive polynomials

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- by P. J. O’Hara and R. S. Rodriguez
- Proc. Amer. Math. Soc.
**44**(1974), 331-335 - DOI: https://doi.org/10.1090/S0002-9939-1974-0349967-5
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## 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].

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## Bibliographic Information

- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**44**(1974), 331-335 - MSC: Primary 30A06
- DOI: https://doi.org/10.1090/S0002-9939-1974-0349967-5
- MathSciNet review: 0349967