Balayage of measures and subharmonic functions on a system of rays. III. Growth of entire functions of exponential type

Abstract

In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($\delta$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex plane $\mathbb C$. In this third part of our work, we use only the case $q=1$ when $S$ is a pair of oppositely directed rays, i.e., $S$ is a straight line as the point set, and balayage is made from both sides of this line. We consider measures and subharmonic functions of finite type of order $1$. This bilateral balayage of genus $1$ will be applied to the non-triviality of weight classes of entire functions of exponential type $E$ that allocated only constraint on their growth along the line; for the full description of subsequences of zeros for classes $E$; the existence of entire functions-multipliers of $h$ for entire functions of exponential type $g$, limiting by the multiplication of their growth as $fh\in E$; the possibility of the representation of meromorphic functions in the form of a ratio of functions from $E$. The origins of the study lies in the classical Malliavin-Rubel Theorem on the conditions of existence of an entire function of exponential type vanishing on a given sequence of positive numbers. These studies also are parallel to the famous Beurling-Malliavin Theorems on the multiplier and on the radius of completeness.