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 Probl. Anal. Issues Anal., 2019, Volume 8(26), Issue 2, Pages 67–72 (Mi pa264)

A lower bound for the $L_2[-1, 1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle

M. A. Komarov

Abstract: Let $C$ be the unit circle $ż:|z|=1\}$ and $Q_n(z)$ be an arbitrary $C$-polynomial (i.e., all its zeros $z_1,…, z_n\in C$). We prove that the norm of the logarithmic derivative $Q_n'/Q_n$ in the complex space $L_2[-1, 1]$ is greater than $1/8$.

Keywords: logarithmic derivative, $C$-polynomial, simplest fraction, norm, unit circle.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-31-00312 mol_a This work was supported by RFBR project 18-31-00312 mol_a.

DOI: https://doi.org/10.15393/j3.art.2019.6030

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Bibliographic databases:

UDC: 517.538.5
MSC: 41A20, 41A29
Revised: 20.05.2019
Accepted:20.05.2019
Language:

Citation: M. A. Komarov, “A lower bound for the $L_2[-1, 1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle”, Probl. Anal. Issues Anal., 8(26):2 (2019), 67–72

Citation in format AMSBIB
\Bibitem{Kom19} \by M.~A.~Komarov \paper A lower bound for the $L_2[-1,\,1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle \jour Probl. Anal. Issues Anal. \yr 2019 \vol 8(26) \issue 2 \pages 67--72 \mathnet{http://mi.mathnet.ru/pa264} \crossref{https://doi.org/10.15393/j3.art.2019.6030} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000471801400005}