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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An extremal problem for the geometric mean of polynomials


Authors: E. Beller and D. J. Newman
Journal: Proc. Amer. Math. Soc. 39 (1973), 313-317
MSC: Primary 30A06
DOI: https://doi.org/10.1090/S0002-9939-1973-0316686-X
MathSciNet review: 0316686
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Abstract: Let $ {M_{0,n}}$ be the maximum of the geometric mean of all $ n$th degree polynomials $ {\sum ^n}{a_k}{e^{ikt}}$ which satisfy $ \vert{a_k}\vert = 1,k = 0,1, \cdots ,n$. We show the existence of certain polynomials $ {R_n}$ whose geometric mean is asymptotic to $ \surd n$, thus proving that $ {M_{0,n}}$ is itself asymptotic to $ \surd n$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1973-0316686-X
Keywords: Extremal polynomials, coefficients of constant modulus, geometric mean, asymptotic to mean square, zeros of polynomials, reverse arithmetic-geometric inequality
Article copyright: © Copyright 1973 American Mathematical Society

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