TY - JOUR
T1 - A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions
AU - A. I. Kheyfits
JO - Analysis in Theory and Applications
VL - 1
SP - 136
EP - 140
PY - 2014
DA - 2014/03
SN - 30
DO - http://doi.org/10.4208/ata.2014.v30.n1.10
UR - https://global-sci.org/intro/article_detail/ata/4479.html
KW - Valiron-Titchmarsh theorem, Tauberian theorems for entire functions with negative zeros, Subharmonic functions in $\mathbb{R}^n$ with Riesz masses on a ray, associated Legendre functions on the cut.
AB - <p style="text-align: justify;">The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in $\mathbb{R}^n$, $n\geq 3$. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if $u$ is a subharmonic function of this class and of order $0&lt;\rho&lt;1$, then the existence of the limit $\lim_{r \to \infty} \log u(r)/N(r),$ where $N(r)$ is the integrated counting function of the masses of $u$, implies the regular asymptotic behavior for both $u$ and its associated measure.</p>