On support points of univalent functions and a disproof of a conjecture of Bombieri

Greiner, Richard A.; Roth, Oliver

doi:10.1090/s0002-9939-01-05994-9

Cited by 15 publications

(15 citation statements)

References 11 publications

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“…For (m, n) = (3, 2), this property has been proved by Greiner and Roth in [49]. Such interpretation can serve to define a number σ mn (M ), M > 1, as a maximal number for which Re (a n − λa m ) is locally maximized on S(M ) by the Pick function P M (z) = M K −1 (K(z)/M ).…”

Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 86%

“…Greiner and Roth [49] answered negatively to partial Bombieri's conjecture. They evaluated σ 32 showing that σ 32 < B 32 .…”

Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 99%

“…We see that Bombieri's conjecture is attacked by different methods: Schiffer's differential equation for support points of S [48,49], variational method in the class S [52, [55][56][57], Lebedev -Milin inequalities based on the Grunsky approach [54], the Loewner theory and optimal control methods and the Pontryagin maximum principle on solutions to the Loewner ODE [47,50,51,53,58]. Roth [27] gave a statement of Teichmüller's coefficient theorem entirely in terms of the Loewner equation.…”

Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 99%

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Value Regions in Classes of Conformal Mappings

Prokhorov¹

2019

MMI

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The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular. A value region is the set {f (z 0 )} of all possible values for the functional f → f (z 0 ) where z 0 is a fixed point either in the upper half-plane for the chordal case or in the unit disk for the radial case, and f runs through a class of conformal mappings. Solutions to the Loewner differential equations form dense subclasses of function families under consideration. The coefficient value regions {(a 2 , . . . , a n ) : f (z) = z + ∞ n=2 a n z n }, |z| < 1, are the part of the field closely linked with extremal problems and the Bombieri conjecture about the structure of the coefficient region for the class S in a neighborhood of the point (2, . . . , n) corresponding to the Koebe function.Благодарности. Исследование выполнено при финансовой поддержке Российского научного фонда (проект № 17-11-01229). Образец для цитирования:Prokhorov D. V. Value Regions in Classes of Conformal Mappings [Прохоров Д. В. Области значений в классах конформных отображений] // Изв. Сарат. ун-та. Нов. сер.

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Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 86%

“…Greiner and Roth [49] answered negatively to partial Bombieri's conjecture. They evaluated σ 32 showing that σ 32 < B 32 .…”

Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 99%

Section: ✷✼✷ ❮àó÷íûé îòäåëmentioning

confidence: 99%

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Value Regions in Classes of Conformal Mappings

Prokhorov¹

2019

MMI

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“…It is noteworthy that D. Bshouty and W. Hengartner [7] proved Bombieri's conjecture for functions from S having real coefficients in their Taylor's expansion. Continuing this contribution by D. Bshouty and W. Hengartner, the conjecture for the whole class S has been recently disproved by R. Greiner and O. Roth [9] for n = 2, m = 3, f ∈ S. Actually, they have got the sharp Bombieri number σ 32 = (e − 1)/4e < 1/4 = B 32 .…”

mentioning

confidence: 89%

Optimal Control in Bombieri's and Tammi's Conjectures

Prokhorov

Vasil'ev

2005

GMJ

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Let 𝑆 stand for the usual class of univalent regular functions in the unit disk 𝑈 = {𝑧 : |𝑧| < 1} normalized by 𝑓(𝑧) = 𝑧 + 𝑎2𝑧2 + . . . in 𝑈, and let 𝑆𝑀 be its subclass defined by restricting |𝑓(𝑧)| < 𝑀 in 𝑈, 𝑀 ≥ 1. We consider two classical problems: Bombieri's coefficient problem for the class 𝑆 and the sharp estimate of the fourth coefficient of a function from 𝑆𝑀. Using Löwner's parametric representation and the optimal control method we propose a way to finding initial Bombieri's numbers and derive a sharp constant 𝑀0 such that for all 𝑀 ≥ 𝑀0 the Pick function gives the local maximum to |𝑎4|. Numerical approximation is given.

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“…where the lim inf are taken over all functions f (z) = z + ∞ j=2 a j z j ∈ S. Recently, this conjecture has been disproved for n = 2 and m = 3 (cf. [13]) while the other cases are still open. We shall first show how the methods of Sections 1 and 2 can be used to give a short proof of:…”

mentioning

confidence: 95%