Geodesic
Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions
Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 444-452 
P. V. Filevich. Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions. Matematičeskie zametki, Tome 75 (2004) no. 3, pp. 444-452. http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a11/
@article{MZM_2004_75_3_a11,
     author = {P. V. Filevich},
     title = {Asymptotic {Relations} between {Maximums} of {Absolute} {Values} and {Maximums} of {Real} {Parts} of {Entire} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {444--452},
     year = {2004},
     volume = {75},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a11/}
}
TY  - JOUR
AU  - P. V. Filevich
TI  - Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions
JO  - Matematičeskie zametki
PY  - 2004
SP  - 444
EP  - 452
VL  - 75
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a11/
LA  - ru
ID  - MZM_2004_75_3_a11
ER  - 
%0 Journal Article
%A P. V. Filevich
%T Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions
%J Matematičeskie zametki
%D 2004
%P 444-452
%V 75
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2004_75_3_a11/
%G ru
%F MZM_2004_75_3_a11

Voir la notice de l'article provenant de la source Math-Net.Ru

According to the classical Wiman–Valiron theorem, the maximum of the absolute value and the maximum of the real part of an entire function are asymptotically equal at infinity outside an exceptional set of a finite logarithmic measure. In this paper, we study the following problem concerning the exceptional set: how does the ratio of the maximum of the absolute value and the maximum of the real part of an entire function depend on its Taylor coefficients? In particular, our results imply that the maximum of the absolute value can increase arbitrarily fast with respect to the maximum of the real part or the Nevanlinna characteristic.

[1] Sheremeta M. N., “O sootnosheniyakh mezhdu maksimalnym chlenom i maksimumom modulya tselogo ryada Dirikhle”, Matem. zametki, 51:5 (1992), 141–148 | MR | Zbl

[2] Valiron G., Lectures on the general theory of integral functions, Édouard Privat, Toulouse, 1923

[3] Vittikh G., Noveishie issledovaniya po odnoznachnym analiticheskim funktsiyam, Fizmatgiz, M., 1960 | MR

[4] Bergweiler W., “On meromorphic funtions that share three values and on the exeptional set in Wiman–Valiron theory”, Kodai Math. J., 13:1 (1990), 1–9 | DOI | MR | Zbl

[5] Skaskiv O. B., Filevich P. V., “Pro velichinu vinyatkovoï mnozhini v teoremi Vimana”, Matem. Studiï, 12:1 (1999), 31–36 | MR | Zbl

[6] Kheiman U. K., Meromorfnye funktsii, Mir, M., 1966 | MR

[7] Strelits Sh. I., Asimptoticheskie svoistva analiticheskikh reshenii differentsialnykh uravnenii, Mintis, Vilnyus, 1972 | MR | Zbl

[8] Polia G., Sege G., Zadachi i teoremy iz analiza, T. 2, Nauka, M., 1978