Geodesic
Entire functions with preassigned zero proximate order
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 141-153 
A. F. Grishin; Nguyen Van Quynh. Entire functions with preassigned zero proximate order. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 42, Tome 424 (2014), pp. 141-153. http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/
@article{ZNSL_2014_424_a3,
     author = {A. F. Grishin and Nguyen Van Quynh},
     title = {Entire functions with preassigned zero proximate order},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {141--153},
     year = {2014},
     volume = {424},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/}
}
TY  - JOUR
AU  - A. F. Grishin
AU  - Nguyen Van Quynh
TI  - Entire functions with preassigned zero proximate order
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2014
SP  - 141
EP  - 153
VL  - 424
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/
LA  - ru
ID  - ZNSL_2014_424_a3
ER  - 
%0 Journal Article
%A A. F. Grishin
%A Nguyen Van Quynh
%T Entire functions with preassigned zero proximate order
%J Zapiski Nauchnykh Seminarov POMI
%D 2014
%P 141-153
%V 424
%U http://geodesic.mathdoc.fr/item/ZNSL_2014_424_a3/
%G ru
%F ZNSL_2014_424_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

It is known that if the proximate order $\rho(r)$ is such that $\lim\rho(r)=\rho>0$ ($r\to\infty$), then there exists an entire function $f(z)$ of proximate order $\rho(r)$. In the case where $\rho=0$ the question about the existence of such an entire function has remained open until now. This question is investigated in the paper.

[1] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[2] A. F. Grishin, I. V. Poedintseva, “Abelevy i tauberovy teoremy dlya integralov”, Algebra i analiz, 26:3 (2014), 1–88

[3] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Cambridge University Press, Cambridge–London–New York–New Rochelle–Melbaurne–Sidney, 1987 | MR | Zbl

[4] A. F. Grishin, T. I. Malyutina, “Ob utochnënnom poryadke”, Kompleksnyi analiz i matematicheskaya fizika, Sbornik, Krasnoyarsk, 1998, 10–24

[5] A. F. Grishin, A. Shuigi, “K teorii predelnykh mnozhestv Azarina”, Matematichni Studiï, 28:2 (2007), 163–174 | MR

[6] R. S. Yulmukhametov, “Approksimatsiya subgarmonicheskikh funktsii”, Analysis Mathematica, 11:3 (1985), 257–282 | DOI | MR | Zbl