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An inequality for derivatives of polynomials whose zeros are in a half-plane

Author: Faruk F. Abi-Khuzam
Journal: Proc. Amer. Math. Soc. 89 (1983), 119-124
MSC: Primary 30C10; Secondary 30D20
MathSciNet review: 706523
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Abstract: Let $ Q$ be a real polynomial of degree $ N$ all of whose zeros lie in the half-plane $ \operatorname{Re} z \leqslant 0$. Let $ M(r,Q)$ be the maximum of $ \left\vert {Q(z)} \right\vert{\text{on}}\left\vert z \right\vert = r$ and $ n(r,0)$ the counting function of the zeros of $ Q$. It is shown that the inequality $ M(r,Q') \leqslant {(2r)^{ - 1}}\left\{ {N + n(r,0)} \right\}M(r,Q)$ holds for $ r > 0$. It is also shown that Bernstein's inequality characterizes polynomials.

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

  • [1] Abdul Aziz and Q. G. Mohammad, Simple proof of a theorem of Erdös and Lax, Proc. Amer. Math. Soc. 80 (1980), 119-122. MR 574519 (81g:30008)
  • [2] J. E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, 1944. MR 0012121 (6:261f)
  • [3] E. C. Titchmarsh, The theory of functions (2nd ed.), Oxford Univ. Press, 1939. MR 0197687 (33:5850)

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Article copyright: © Copyright 1983 American Mathematical Society

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