A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions

Abstract

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.