no. 8
Complex Analysis/Mathematical Analysis
Maximum modulus points and zero sets of entire functions of regular growth
[Points de module maximal et ensemble de zéros des fonctions entières de croissance regulière]

Présenté par : Jean-Pierre Kahane

Ostrovskii, Iossif1, 2 ; Üreyen, Ersin1
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, Tome 341 (2005) no. 8, pp. 481-484.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2005.09.012
Ostrovskii, Iossif 1, 2 ; Üreyen, Ersin 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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Ostrovskii, Iossif; Üreyen, Ersin. Maximum modulus points and zero sets of entire functions of regular growth. Comptes Rendus. Mathématique, Tome 341 (2005) no. 8, pp. 481-484. doi : 10.1016/j.crma.2005.09.012. http://www.numdam.org/articles/10.1016/j.crma.2005.09.012/

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

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

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

[5] Ostrovskii, I.V.; Üreyen, A.E. 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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