An Extremal Problem for Odd Univalent Polynomials

Dmitriy Dmitrishin; Daniel Gray; Alexander Stokolos; Iryna Tarasenko

doi:10.1007/s40315-023-00487-3

An Extremal Problem for Odd Univalent Polynomials

Dmitriy Dmitrishin, Daniel Gray, Alexander Stokolos, Iryna Tarasenko

Mathematical Sciences

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

2 Scopus citations

Abstract

For the univalent polynomials F(z)=∑_j=1^Na_jz^2j-1 with real coefficients and normalization a₁=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where U_j(x) is a Chebyshev polynomial of the second kind and U_j^′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.

Original language	English
Pages (from-to)	83-100
Number of pages	18
Journal	Computational Methods and Function Theory
Volume	24
Issue number	1
DOIs	https://doi.org/10.1007/s40315-023-00487-3
State	Published - Mar 2024

Scopus Subject Areas

Analysis
Computational Theory and Mathematics
Applied Mathematics

Keywords

Chebyshev polynomials
Koebe one-quarter theorem
Odd univalent polynomials
T-folded Koebe function

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10.1007/s40315-023-00487-3

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title = "An Extremal Problem for Odd Univalent Polynomials",

abstract = "For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.",

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AU - Dmitrishin, Dmitriy

AU - Gray, Daniel

AU - Stokolos, Alexander

AU - Tarasenko, Iryna

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.

PY - 2024/3

Y1 - 2024/3

N2 - For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.

AB - For the univalent polynomials F(z)=∑j=1Najz2j-1 with real coefficients and normalization a1=1 we solve the extremal problem (Formula presented.) We show that the solution is (Formula presented.) and the extremal polynomial (Formula presented.) is unique and univalent, where Uj(x) is a Chebyshev polynomial of the second kind and Uj′(x) denotes the derivative. As an application, we obtain an estimate of the Koebe radius for odd univalent polynomials in D and formulate several conjectures.

KW - Chebyshev polynomials

KW - Koebe one-quarter theorem

KW - Odd univalent polynomials

KW - T-folded Koebe function

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An Extremal Problem for Odd Univalent Polynomials

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