Geodesic
Exactness of approximation of a subharmonic function by the logarithm of the modulus of an analytic function in the Chebyshev metric
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 55-73 
M. A. Hirnyk. Exactness of approximation of a subharmonic function by the logarithm of the modulus of an analytic function in the Chebyshev metric. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 55-73. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a3/
@article{ZNSL_2005_327_a3,
     author = {M. A. Hirnyk},
     title = {Exactness of approximation of a~subharmonic function by the logarithm of the modulus of an analytic function in the {Chebyshev} metric},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {55--73},
     year = {2005},
     volume = {327},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a3/}
}
TY  - JOUR
AU  - M. A. Hirnyk
TI  - Exactness of approximation of a subharmonic function by the logarithm of the modulus of an analytic function in the Chebyshev metric
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2005
SP  - 55
EP  - 73
VL  - 327
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a3/
LA  - ru
ID  - ZNSL_2005_327_a3
ER  - 
%0 Journal Article
%A M. A. Hirnyk
%T Exactness of approximation of a subharmonic function by the logarithm of the modulus of an analytic function in the Chebyshev metric
%J Zapiski Nauchnykh Seminarov POMI
%D 2005
%P 55-73
%V 327
%U http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a3/
%G ru
%F ZNSL_2005_327_a3

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

It is known that a subharmonic function $u(z)$ of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function $f(z)$ at the point $z$ up to $C\log|z|$ outside a very small exceptional set. We prove that if a constant $C$ decreases, then, beginning with the value $C=\rho/4$, the exceptional set enlarges substantially. This improves a result by Yulmukhametov. We also prove similar results for subharmonic functions of infinite order and functions subharmonic in the disk. The main result of the article is the following. Theorem 1. Suppose a number $\rho$ is positive, and an entire function $f(z)$ satisfies the condition $$ ||z|^\rho-\log|f(z)||\le C\log|z|, \qquad z\notin E, $$ where $E\subset\bigcup_j\{z:|z-z_j|, $r_j<|z_j|^{1-\rho/2-2C+\varepsilon}$, and $\varepsilon>0$. Then $$ \sum_{R\le|z_j|\le 2R}r_j\ge R^{1+\rho/2-2C-3\varepsilon}, \qquad R>R(\varepsilon). $$

[1] S. N. Mergelyan, “Ravnomernye priblizheniya funktsii kompleksnogo peremennogo”, UMN, 7:2 (1952), 31–122 | MR | Zbl

[2] A. Beurling, P. Malliavin, “On Fourier transforms of measures with compact support”, Acta Math., 107 (1962), 291–391 | DOI | MR

[3] N. U. Arakelyan, “Tselye funktsii konechnogo poryadka s beskonechnym mnozhestvom defektnykh znachenii”, Dokl. AN SSSR, 170:2 (1966), 999–1002 | Zbl

[4] P. Z. Agranovich, V. N. Logvinenko, O massivnosti isklyuchitelnogo mnozhestva mnogochlennogo asimptoticheskogo predstavleniya subgarmonicheskoi funktsii, Preprint No 45–87, FTINT AN USSR, Kharkov, 1988

[5] Yu. I. Lyubarskii, M. L. Sodin, Analogi funktsii tipa sinusa dlya vypuklykh oblastei, Preprint No 17–86, FTINT AN USSR, Kharkov, 1986

[6] D. Drasin, “Approximation of subharmonic functions with applications”, Approximation, Complex Analysis, and Potential Theory, Proc. of the NATO ASI. (Montreal, Canada, 3 to 14 July 2000), Kluwer Academic Publishers, Dordrecht–Boston–London, 2001, 163–189 | MR | Zbl

[7] B. Kielberg, On certain integral and harmonic functions. A study in minimum modulus, Thesis, University of Uppsala, 1948 | MR

[8] P. B. Kennedy, “A class of integral functions bounded on certain curves”, Proc. London Math. Soc., 6 (1956), 518–547 | DOI | MR | Zbl

[9] W. Al-Katifi, “On the asymptotic values and parths of certain integral and meromorphic functions”, Proc. London Math. Soc., 16 (1966), 599–634 | DOI | MR | Zbl

[10] V. S. Azarin, “O luchakh vpolne regulyarnogo rosta tseloi funktsii”, Mat. sb., 79:4 (1969), 463–476 | MR | Zbl

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

[12] M. Girnyk, A. Goldberg, “Approximation of subharmonic functions by logarithms of moduli of entire functions in integral metrics”, Israel Math. Conf. Proc., 15 (2001), 117–135 | MR | Zbl

[13] Yu. Lyubarskii, Eu. Malinnikova, “On approximation of subharmonic functions”, J. d'Analyse Math., 83 (2001), 121–149 | DOI | MR | Zbl

[14] I. Chyzhykov, “Approximation of subharmonic functions of slow growth”, Math. Fiz. Anal. Geom., 9 (2002), 509–520 | MR | Zbl

[15] I. Chizhikov, “Approksimatsiya subgarmonicheskikh funktsii”, Algebra i Analiz, 16:3 (2004), 211–237 | MR | Zbl

[16] W. K. Hayman, P. B. Kennedy, Subharmonic functions, V. 1, Academic Press, London–New York–San Francisco, 1976 | MR

[17] M. A. Girnyk, “O priblizhenii subgarmonicheskoi funktsii beskonechnogo poryadka logarifmom modulya tseloi funktsii”, Mat. zametki, 50:4 (1991), 57–60 | MR

[18] M. A. Girnyk, “Priblizhenie funktsii, subgarmonicheskoi v kruge, logarifmom modulya analiticheskoi funktsii”, Ukr. mat. zh., 48:8 (1994), 1080–1083 | MR

[19] A. A. Goldberg, I. V. Ostrovskii, Raspredelenie znachenii meromorfnykh funktsii, Nauka, M., 1970 | MR

[20] N. S. Landkof, Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl