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

Iossif Ostrovskii; Ersin Üreyen

doi:10.1016/j.crma.2005.09.012

Complex Analysis/Mathematical Analysis

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

Presented by: Jean-Pierre Kahane

Iossif Ostrovskii ^1,²; Ersin Üreyen ¹

¹ Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
² B.Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkov, Ukraine

Comptes Rendus. Mathématique, Volume 341 (2005) no. 8, pp. 481-484.

Abstract
Résumé

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.

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.

Received: 2005-03-01
Accepted: 2005-09-01
Published online: 2005-10-05

DOI: 10.1016/j.crma.2005.09.012

Author's affiliations:

Iossif Ostrovskii ^{1, 2}; Ersin Üreyen ¹

¹ Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
² B.Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkov, Ukraine

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     title = {Maximum modulus points and zero sets of entire functions of regular growth},
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     pages = {481--484},
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     number = {8},
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     doi = {10.1016/j.crma.2005.09.012},
     language = {en},
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Iossif Ostrovskii; Ersin Üreyen. Maximum modulus points and zero sets of entire functions of regular growth. Comptes Rendus. Mathématique, Volume 341 (2005) no. 8, pp. 481-484. doi : 10.1016/j.crma.2005.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.012/

References
Cited by

[1] J. Clunie; T. Kövari On integral functions having prescribed asymptotic growth. II, Canad. J. Math., Volume 20 (1968), pp. 7-20

[2] J.P. Earl; W.K. Hayman Smooth majorants for functions of arbitrarily rapid growth, Math. Proc. Cambridge Philos. Soc., Volume 109 (1991), pp. 565-569

[3] B.Ya. Levin Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1980

[4] A.J. Macintyre Wiman's method and “flat regions” of integral functions, Quart. J. Math., Volume 9 (1938), pp. 81-88

[5] I.V. Ostrovskii; A.E. Üreyen Distance between a maximum modulus point and zero set of an entire function, Complex Variables, Volume 48 (2003), pp. 583-598

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