|
This article is cited in 7 scientific papers (total in 7 papers)
Turán–Erőd type converse Markov inequalities on general convex domains of the plane in the boundary $L^q$ norm
P. Yu. Glazyrinaab, Sz. Gy. Révészc a Institute of Natural Sciences and Mathematics, Ural Federal University named after the First President of Russia B. N. Yeltsin, ul. Kuibysheva 48, Yekaterinburg, 620026 Russia
b N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990 Russia
c Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, Budapest, 1053 Hungary
Abstract:
In 1939 P. Turán started to derive lower estimations on the norm of the derivatives of polynomials of (maximum) norm $1$ on $\mathbb I:=[-1,1]$ (interval) and $\mathbb D:=\{z\in \mathbb C: |z|\le 1\}$ (disk) under the normalization condition that the zeroes of the polynomial in question all lie in $\mathbb I$ or $\mathbb D$, respectively. For the maximum norm he found that with $n:=\deg p$ tending to infinity, the precise growth order of the minimal possible derivative norm is $\sqrt {n}$ for $\mathbb I$ and $n$ for $\mathbb D$. J. Erőd continued the work of Turán considering other domains. Finally, about a decade ago the growth of the minimal possible $\infty $-norm of the derivative was proved to be of order $n$ for all compact convex domains. Although Turán himself gave comments about the above oscillation question in $L^q$ norms, till recently results were known only for $\mathbb D$ and $\mathbb I$. Recently, we have found order $n$ lower estimations for several general classes of compact convex domains, and conjectured that even for arbitrary convex domains the growth order of this quantity should be $n$. Now we prove that in $L^q$ norm the oscillation order is at least $n/\kern -1pt\log n$ for all compact convex domains.
Keywords:
Bernstein–Markov inequalities, Turán's lower estimate of derivative norm, logarithmic derivative, convex domains, Chebyshev constant, transfinite diameter, capacity, minimal width, outer angle.
Received: May 10, 2018
Citation:
P. Yu. Glazyrina, Sz. Gy. Révész, “Turán–Erőd type converse Markov inequalities on general convex domains of the plane in the boundary $L^q$ norm”, Harmonic analysis, approximation theory, and number theory, Collected papers. Dedicated to Academician Sergei Vladimirovich Konyagin on the occasion of his 60th birthday, Trudy Mat. Inst. Steklova, 303, MAIK Nauka/Interperiodica, Moscow, 2018, 87–115; Proc. Steklov Inst. Math., 303 (2018), 78–104
Linking options:
https://www.mathnet.ru/eng/tm3957https://doi.org/10.1134/S0371968518040088 https://www.mathnet.ru/eng/tm/v303/p87
|
Statistics & downloads: |
Abstract page: | 330 | Full-text PDF : | 62 | References: | 52 | First page: | 10 |
|