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Izv. Vyssh. Uchebn. Zaved. Mat., 2010, Number 3, Pages 92–96 (Mi ivm6716)  

This article is cited in 1 scientific paper (total in 1 paper)

Brief communications

Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

A. V. Parfenenkov

Chair of Mathematical Analysis and Function Theory, Ural State University, Ekaterinburg, Russia

Abstract: We consider the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable with complex coefficients of degree at most $n$ with real constant terms. In this class we estimate the uniform norm of a polynomial $P_n\in\mathcal P_n^*$ on the circle $\Gamma_r=ż\in\mathbb C\colon|z|=r\}$ of radius $r>1$ in terms of the norm of its real part on the unit circle $\Gamma_1$. More precisely, we study the best constant $\mu(r,n)$ in the inequality $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. We prove that $\mu(r,n)=r^n$ for $r^{n+2}-r^n-3r^2-4r+1\geq0$. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality $\mu(r,n)>r^n$ is valid for $r$ sufficiently close to 1.

Keywords: inequalities for algebraic polynomials, uniform norm, circle in complex plane.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2010, 54:3, 80–83

Bibliographic databases:

UDC: 517.518
Received: 19.06.2009

Citation: A. V. Parfenenkov, “Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle”, Izv. Vyssh. Uchebn. Zaved. Mat., 2010, no. 3, 92–96; Russian Math. (Iz. VUZ), 54:3 (2010), 80–83

Citation in format AMSBIB
\Bibitem{Par10}
\by A.~V.~Parfenenkov
\paper Estimation of an algebraic polynomial in a~plane in terms of its real part on the unit circle
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2010
\issue 3
\pages 92--96
\mathnet{http://mi.mathnet.ru/ivm6716}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2778329}
\transl
\jour Russian Math. (Iz. VUZ)
\yr 2010
\vol 54
\issue 3
\pages 80--83
\crossref{https://doi.org/10.3103/S1066369X10030126}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78649549065}


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    This publication is cited in the following articles:
    1. A. V. Parfenenkov, “Tochnoe neravenstvo mezhdu ravnomernymi normami algebraicheskogo mnogochlena i ego veschestvennoi chasti na kontsentricheskikh okruzhnostyakh kompleksnoi ploskosti”, Tr. IMM UrO RAN, 16, no. 4, 2010, 254–263  mathnet  elib
  • Известия высших учебных заведений. Математика Russian Mathematics (Izvestiya VUZ. Matematika)
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