On the Malliavin-Rubel theorem on small entire functions of exponential type with given zeros

TL;DR

This paper extends the Malliavin-Rubel theorem to more general distributions of complex zeros outside certain angular regions, broadening the understanding of entire functions of exponential type with prescribed zeros.

Contribution

It generalizes the classical Malliavin-Rubel theorem to complex zero distributions outside specific angular sectors.

Findings

Extended the theorem to complex zeros outside vertical angles

Provided conditions for the existence of entire functions with prescribed zeros

Broadened the class of zero distributions for exponential type functions

Abstract

In the early 1960s, P. Malliavin and L. A. Rubel gave a complete description of pairs of distributions of positive points $Z$ and $W$ such that for each entire function of exponential type $g\neq 0$ that vanishes on $W$, there is an entire function of exponential type $f\neq 0$ such that $f$ vanishes on $Z$ and satisfies the inequality $|f|\leq |g|$ everywhere on the imaginary axis. We extend this result to much more general distributions of complex points $Z$ and $W$ lying outside of some pair of vertical angles containing the imaginary axis as the points approach $\infty$.