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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 inequalities of algebraic polynomials


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 123 (1995), 2041-2048
MSC: Primary 26D05; Secondary 41A10
MathSciNet review: 1231305
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Abstract: Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by $ {H_n}$ the set of all polynomials whose degree is n and whose zeros are real and lie inside $ [ - 1,1)$. Let $ {P_n} \in {H_n}$ and $ {P_n}(1) = 1$; then the object of Theorem 1 is to obtain the best lower bound of the expression $ \smallint _{ - 1}^1\vert P_n'(x){\vert^p}\,dx$ for $ p \geq 1$ and characterize the polynomial which achieves this lower bound. Next, we say that $ {P_n} \in {S_n}[0,\infty )$ if $ {P_n}$ is a polynomial whose degree is n and whose roots are all real and do not lie inside $ [0,\infty )$. In Theorem 2, we shall prove Markov-type inequality for such a class of polynomials belonging to $ {S_n}[0,\infty )$ in the weighted $ {L_p}$ norm (p integer). Here $ {\left\Vert {{P_n}} \right\Vert _{{L_p}}} = {(\smallint _0^\infty \vert{P_n}(x){\vert^p}{e^{ - x}}\,dx)^{1/p}}$. In Theorem 3 we shall consider another analogous problem as in Theorem 2.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1231305-0
PII: S 0002-9939(1995)1231305-0
Article copyright: © Copyright 1995 American Mathematical Society