TY - JOUR T1 - A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions 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 -

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 $0<\rho<1$, 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.