Abstract:
Let $C$ be the unit circle $\{z:|z|=1\}$ and $Q_n(z)$ be an arbitrary $C$-polynomial (i.e., all its zeros $z_1,\dots, z_n\in C$).
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.
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
\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}
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\elib{https://elibrary.ru/item.asp?id=38224562}
Linking options:
https://www.mathnet.ru/eng/pa264
https://www.mathnet.ru/eng/pa/v26/i2/p67
This publication is cited in the following 4 articles:
P. A. Borodin, A. M. Ershov, “S. R. Nasyrov's Problem of Approximation by Simple Partial Fractions on an Interval”, Math. Notes, 115:4 (2024), 520–527
M. A. Komarov, “Density of Simple Partial Fractions with Poles on a Circle in Weighted Spaces for a Disk and a Segment”, Vestnik St.Petersb. Univ.Math., 57:1 (2024), 62
P. A. Borodin, K. S. Shklyaev, “Density of quantized approximations”, Russian Math. Surveys, 78:5 (2023), 797–851
Mikhail A. Komarov, “A Newman type bound for $L_p[-1,1]$-means of the logarithmic derivative of polynomials having all zeros on the unit circle”, Constr Approx, 58:3 (2023), 551
Citation in format AMSBIB
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