Extremal problems for typically real odd polynomials

D. Dmitrishin; D. Gray; A. Stokolos; I. Tarasenko

doi:10.1007/s10474-024-01440-z

Extremal problems for typically real odd polynomials

D. Dmitrishin, D. Gray, A. Stokolos, I. Tarasenko

Mathematical Sciences

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

Abstract

On the class of typically real odd polynomials of degree 2N-1 (Formula presented.) we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a₂.It is shown that (Formula presented.) and (Formula presented.) where ν_N is a minimal positive root of the equation U_N+1^′(x)=0 with U_N+1^′(x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order.The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.

Original language	English
Pages (from-to)	1-19
Number of pages	19
Journal	Acta Mathematica Hungarica
Volume	173
Issue number	1
DOIs	https://doi.org/10.1007/s10474-024-01440-z
State	Published - Jun 2024

Scopus Subject Areas

General Mathematics

Keywords

30C10
30C25
30C55
30C75
Chebyshev polynomial
extremal problem for polynomials
typically real odd polynomial

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10.1007/s10474-024-01440-z

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abstract = "On the class of typically real odd polynomials of degree 2N-1 (Formula presented.) we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a2.It is shown that (Formula presented.) and (Formula presented.) where νN is a minimal positive root of the equation UN+1′(x)=0 with UN+1′(x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order.The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.",

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AU - Dmitrishin, D.

AU - Gray, D.

AU - Stokolos, A.

AU - Tarasenko, I.

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Akadémiai Kiadó, Budapest, Hungary 2024.

PY - 2024/6

Y1 - 2024/6

N2 - On the class of typically real odd polynomials of degree 2N-1 (Formula presented.) we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a2.It is shown that (Formula presented.) and (Formula presented.) where νN is a minimal positive root of the equation UN+1′(x)=0 with UN+1′(x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order.The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.

AB - On the class of typically real odd polynomials of degree 2N-1 (Formula presented.) we consider two problems: 1) stretching the central unit disc under the above polynomial mappings and 2) estimating the coefficient a2.It is shown that (Formula presented.) and (Formula presented.) where νN is a minimal positive root of the equation UN+1′(x)=0 with UN+1′(x) being the derivative of the Chebyshev polynomial of the second kind of the corresponding order.The above boundaries are sharp, the corresponding estremizers are unique and the coefficients are determined.

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KW - 30C25

KW - 30C55

KW - 30C75

KW - Chebyshev polynomial

KW - extremal problem for polynomials

KW - typically real odd polynomial

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JO - Acta Mathematica Hungarica

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Extremal problems for typically real odd polynomials

Abstract

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