Balayage of measures and subharmonic functions to a system of rays. II. Balayages of finite genus and growth regularity on a single ray

Abstract:

The classical balayages of measures and subharmonic functions are extended to a system of rays $S$ with common origin on the complex plane $\mathbb {C}$. For an arbitrary subharmonic function $v$ of finite order on $\mathbb {C}$, this allows one to build a $\delta$-subharmonic function on $\mathbb {C}$ that is harmonic outside of $S$, coincides with $v$ on $S$ outside of a polar set, and has the same growth order as $v$. Applications are given to the investigation of the relationship between the growth of an entire function on $S$ and the distribution of its zeros. In the present second part of the project, the results and preliminaries of its first part are used essentially.