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.


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.


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How to Cite
Zabolotskyi M. Asymptotics of $\delta$-subharmonic functions of finite order. Mat. Stud. [Internet]. 2020Dec.25 [cited 2021Nov.28];54(2):188-92. Available from: