An extremal problem for polynomials

Dmitrishin, Dmitriy; Smorodin, Andrey; Stokolos, Alexander M.

doi:10.1016/j.acha.2021.08.008

Cited by 7 publications

(5 citation statements)

References 28 publications

(62 reference statements)

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“…Lemma 5.1 [2]. Let A(t) and B(t) be defined as in the boundary Γ + 3 of the domain of univalence U T .…”

Section: Appendix Auxiliary Resultsmentioning

confidence: 99%

“…Thus, the minimum of the function Φ(a, b, φ) is attained at such values of the coefficients that correspond to the common point of the curves Γ 1 and Γ + 3 . This point determines the coefficients of the generalized Suffridge trinomial [2]…”

Section: Curve γmentioning

confidence: 99%

“…• Next, consider T = 2 α = 1 3 . In this case, on the curve Γ + 3 , the parameters a and b are related by the relationship (a + b) 2…”

Section: Curve γmentioning

confidence: 99%

“…In [2,5,12], there was considered the Koebe problem for polynomials of degree N with real coefficients:…”

Section: Introductionmentioning

confidence: 99%

“…In [4], the Koebe problem was considered for the univalent in D odd polynomials with real coefficients…”

Section: Introductionmentioning

confidence: 99%

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Some extremal problems for trinomials with fold symmetry

2022

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The Koebe problem for univalent polynomials with real coefficients is fully solved only for trinomials, which means that in this case the Koebe radius and the extremal polynomial (extremizer) have been found. The general case remains open, but conjectures have been formulated. The corresponding conjectures have also been hypothesized for univalent polynomials with real coefficients and T -fold rotational symmetry. This paper provides confirmation of these hypotheses for trinomials z +az T +1 +bz 2T +1 . Namely, the Koebe radius is r = 4 cos 2 π(1+T ) 2+3T , and the only extremizer of the Koebe problem is the trinomial

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“…Lemma 5.1 [2]. Let A(t) and B(t) be defined as in the boundary Γ + 3 of the domain of univalence U T .…”

Section: Appendix Auxiliary Resultsmentioning

confidence: 99%

Section: Curve γmentioning

confidence: 99%

“…• Next, consider T = 2 α = 1 3 . In this case, on the curve Γ + 3 , the parameters a and b are related by the relationship (a + b) 2…”

Section: Curve γmentioning

confidence: 99%

“…In [2,5,12], there was considered the Koebe problem for polynomials of degree N with real coefficients:…”

Section: Introductionmentioning

confidence: 99%