Smooth majorants for functions of arbitrarily rapid growth

Earl, J. P.; Hayman, W. K.

doi:10.1017/s0305004100069991

Cited by 9 publications

(5 citation statements)

References 2 publications

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“…naturally follows from results of Earl and Hayman [2]. A function ρ(r) ∈ C 3 (R + ) is called an infinite p.o.…”

Section: Introductionmentioning

confidence: 93%

“…D'après les résultats de Earl et Hayman [2] il est naturel de définir un o.p. infini comme une fonction ρ(r) ∈ C 3 (R + ) représentable dans la forme (9) où ϑ est une fonction positive convexe telle que (10) est satisfait.…”

Section: Version Française Abrégéeunclassified

“…1), Earl et Hayman [2] montrent que la réunion de toutes les classes A ρ , 0 ρ ∞, est une echelle de croissance des fonctions entières dans le sens suivant : pour toute fonction entière…”

Section: Version Française Abrégéeunclassified

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Maximum modulus points and zero sets of entire functions of regular growth

Островскии

Üreyen

2005

Comptes Rendus. Mathématique

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We obtain lower asymptotic at ∞ estimates of the distance between a maximum modulus point and zero set of an entire function provided that the function is of regular growth with respect to a proximate order. The more regular the growth is the better the estimates are, and they are sharp in some sense. The case of infinite order is also considered; in this case a suitable analogue of usual proximate order is exploited. To cite this article: I. Ostrovskii, E. Üreyen, C. R. Acad. Sci. Paris, Ser. I 341 (2005).  2005 Académie des sciences. Published by Elsevier SAS. All rights reserved. RésuméPoints de module maximal et ensemble de zéros des fonctions entières de croissance regulière. Nous obtenons des estimations inférieure asymptotiques à l'infini de la distance entre un point de module maximal et l'ensemble des zéros d'une fonction entière, quand la fonction est supposée de croissance régulière par rapport à un ordre précisé. Les estimations s'améliorent avec la régularité, et dans un sens elles sont précises. Le cas d'ordre infini est aussi consideré ; dans ce cas nous avons utilisé une analogie appropriée de l'ordre précisé habituel. Pour citer cet article : I. Ostrovskii, E. Üreyen, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

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“…naturally follows from results of Earl and Hayman [2]. A function ρ(r) ∈ C 3 (R + ) is called an infinite p.o.…”

Section: Introductionmentioning

confidence: 93%

Section: Version Française Abrégéeunclassified

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Maximum modulus points and zero sets of entire functions of regular growth

Островскии

Üreyen

2005

Comptes Rendus. Mathématique

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“…is well-defined and the quantity lim 𝑟→+∞ ln 𝑀 (𝑓 ; 𝑟) 𝜈(𝑟) is finite and positive. The original statement by Valiron was extended for entire functions of zero and infinite orders, see [16]. Thus, for each entire function 𝑓 of order 𝜌, where 0 ⩽ 𝜌 ⩽ +∞, there exists an entire function 𝜈 obeying condition (1.9) such that 0 < 𝜎 𝜈 (𝑓 ) := lim 𝑟→+∞ ln 𝑀 (𝑓 ; 𝑟) 𝜈(𝑟) < +∞.…”

Section: 3mentioning

confidence: 99%

Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions

Braichev,

Khabibullin,

Sherstyukov

2023

Ufimsk. Mat. Zh.

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The idea to write this note arose during the discussion that followed the report of the first author at the International Scientific Conference "Ufa Autumn Mathematical School-2022". We propose three general methods for constructing uniqueness sets in classes of entire functions with growth restrictions. In all three cases, the sequence of zeros of an entire function with special properties is chosen as such set. The first method is related to Sylvester famous problem on the smallest circle containing a given set of points on a plane, and theorems of convex geometry. The second method initially relies on Helly theorem on the intersection of convex sets and its application to the possibility of covering one set by shifting another. The third method is based on the classical Jensen formula, which allows one to estimate the type of an entire function via the averaged upper density of the sequence of its zeros. We present only basic results now. The development of our approaches is expected to be presented in subsequent works.

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“…' П того, условие б) равносильно соотношению *"(г)<р(г) } SU P <i ^ W/ С ' ) < +°°> г>0 а последнее хорошо известно, что выполняется для довольно широкого класса ма жорант (см. [18]).…”

Section: <ж)unclassified