An extremal problem for polynomials

Dmitriy Dmitrishin; Andrey Smorodin; Alex Stokolos

doi:10.1016/j.acha.2021.08.008

An extremal problem for polynomials

Dmitriy Dmitrishin, Andrey Smorodin, Alex Stokolos

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

For the polynomials F(z)=∑_j=1^Na_jz^j with real coefficients and normalization a₁=1 we solve the extremal problem supa₂,…,a_N⁡(infz∈D⁡{Re(F(z)):Im(F(z))=0}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the U_j(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

Original language	English
Pages (from-to)	283-305
Number of pages	23
Journal	Applied and Computational Harmonic Analysis
Volume	56
DOIs	https://doi.org/10.1016/j.acha.2021.08.008
State	Published - Jan 2022

Scopus Subject Areas

Applied Mathematics

Keywords

Chebyshev polynomials
Extremal problems
Fejér-Riesz representation
Koebe one-quarter theorem
Variational methods

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10.1016/j.acha.2021.08.008

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title = "An extremal problem for polynomials",

abstract = "For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problem supa2,…,aN⁡(infz∈D⁡\{Re(F(z)):Im(F(z))=0\}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.",

keywords = "Chebyshev polynomials, Extremal problems, Fej{\'e}r-Riesz representation, Koebe one-quarter theorem, Variational methods",

author = "Dmitriy Dmitrishin and Andrey Smorodin and Alex Stokolos",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2022",

month = jan,

doi = "10.1016/j.acha.2021.08.008",

language = "English",

volume = "56",

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T1 - An extremal problem for polynomials

AU - Dmitrishin, Dmitriy

AU - Smorodin, Andrey

AU - Stokolos, Alex

PY - 2022/1

Y1 - 2022/1

N2 - For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problem supa2,…,aN⁡(infz∈D⁡{Re(F(z)):Im(F(z))=0}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

AB - For the polynomials F(z)=∑j=1Najzj with real coefficients and normalization a1=1 we solve the extremal problem supa2,…,aN⁡(infz∈D⁡{Re(F(z)):Im(F(z))=0}). We show that the solution is [Formula presented], and the extremal polynomial [Formula presented] is unique and univalent, where the Uj(x) are the Chebyshev polynomials of the second kind, j=1,…,N. As an application, we obtain the estimate of the Koebe radius for the univalent polynomials in D and formulate several conjectures.

KW - Chebyshev polynomials

KW - Extremal problems

KW - Fejér-Riesz representation

KW - Koebe one-quarter theorem

KW - Variational methods

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DO - 10.1016/j.acha.2021.08.008

M3 - Article

AN - SCOPUS:85120084795

SN - 1063-5203

VL - 56

SP - 283

EP - 305

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -

An extremal problem for polynomials

Abstract

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Keywords

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