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Proceedings of the American Mathematical Society

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



Growth of polynomials with zeros outside a circle

Authors: Abdul Aziz and Q. G. Mohammad
Journal: Proc. Amer. Math. Soc. 81 (1981), 549-553
MSC: Primary 30C10; Secondary 26D05
MathSciNet review: 601727
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Abstract: Let $ P(z)$ be a polynomial of degree $ n$ having all its zeros in $ \vert z\vert \geqslant k \geqslant 1$. For $ k = 1$, it is known that

$\displaystyle \max\limits_{\vert z\vert = R > 1} \vert P(Z)\vert \leqslant \frac{R^n + 1}{2} \max\limits_{\vert z\vert = 1} \vert P(z)\vert.$

In this paper we consider the case $ k > 1$ and obtain a sharp result.

References [Enhancements On Off] (What's this?)

  • [1] N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5 (1955), 849-852. MR 0076020 (17:833e)
  • [2] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509-513. MR 0010731 (6:61f)
  • [3] G. Polya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Springer-Verlag, Berlin, 1925.
  • [4] M. Riesz, Über einen Satz des Herra Serge Bernstein, Acta Math. 40 (1916), 337-349. MR 1555142

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Keywords: Growth of maximum modulus, inequalities for polynomials
Article copyright: © Copyright 1981 American Mathematical Society

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