Asymptotics of $\delta$-subharmonic functions of finite order

  • M.V. Zabolotskyi Ivan Franko National University of Lviv
Keywords: δ-subharmonic function; finite order; asymptotic behavior; proximate order; concentration index.

Abstract

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positive
order we found the asymptotical representation of the form
\[
u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,
\]
where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,
$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$
$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.
The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.

References

V.S. Azarin, Subharmonic functions of completely regular growth, Ph.D., Kharkiv, 1963. (in Russian)

M. Brelot, Étude des fonctions sous-harmoniques au voisinage d’un point singulier, Ann. Inst. Fourier,(1949), 121-156. doi:10.5802/aif.11

A.A. Goldberg, N.V. Zabolotskii, Concentration index of a subharmonic function of zero order, Mat. Zametki, 34 (1983), No2, 227–236. (in Russian)

W.K. Hayman, P.B. Kennedy, Subharmonic Functions, Mir, Moscow, 1980. (in Russian)

T.A. Kolomiitseva, On the asymptotic behavior of an entire function with regular distribution of roots, Teor. Funkts., Funktsional. Anal. Prilozh., 15 (1972), 35–43. (in Russian)

I.F. Krasichkov, Lower bounds for entire functions of finite order, Sibirsk. Mat. Zh., 6 (1965), No4, 840–861. (in Russian)

B.Ya. Levin, Distribution of Zeros of Entire Functions, Gostekhizdat., Moscow, 1956. (in Russian)

Published
2020-12-25
How to Cite
1.
Zabolotskyi M. Asymptotics of $\delta$-subharmonic functions of finite order. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Jun.15];54(2):188-92. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/116
Section
Articles