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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some properties of self-inversive polynomials


Authors: P. J. O’Hara and R. S. Rodriguez
Journal: Proc. Amer. Math. Soc. 44 (1974), 331-335
MSC: Primary 30A06
MathSciNet review: 0349967
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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 $ \vert\vert P'\vert\vert = \tfrac{1}{2}n\vert\vert P\vert\vert$ where $ \vert\vert P'\vert\vert$ and $ \vert\vert P\vert\vert$ 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 $ \vert z\vert = 1$ then $ 2\Sigma _{j = 0}^n\vert{a_j}{\vert^2} \leqq \vert\vert P\vert{\vert^2}$. Finally, as a consequence of this inequality, we show that when $ P$ has all its zeros on $ \vert z\vert = 1$ then $ {2^{1/2}}\vert{a_{n/2}}\vert \leqq \vert\vert P\vert\vert$ and $ 2\vert{a_j}\vert \leqq \vert\vert P\vert\vert$ for $ j \ne n/2$. This answers in part a question presented in [3, p. 24].


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0349967-5
PII: S 0002-9939(1974)0349967-5
Keywords: Self-inversive polynomials, coefficients, maximum modulus, inequalities
Article copyright: © Copyright 1974 American Mathematical Society