Decrease in growth of entire and meromorphic functions
B. N. Khabibullin
TL;DR
The work delivers a unified non-asymptotic theory linking subharmonic minorants to the radial growth of entire and meromorphic functions. By proving a central inequality that adds a controlled logarithmic factor to a given subharmonic function via an entire multiplier $h$, the authors derive sharp bounds for multiplying an entire function by $h$, for constructing entire functions vanishing on prescribed point distributions, and for representing meromorphic functions as $f/g$ with controlled growth. The results hinge on Govorov–Petrenko–Dahlberg–Essén-type inequalities and Jensen measures, and are complemented by precise corollaries involving radial counting functions and Nevanlinna characteristics, with Paley-type constants characterizing optimal bounds. The inequalities are non-asymptotic and sharp, providing explicit, universally valid estimates and extremal constructions that have potential applications in complex analysis and value distribution theory.
Abstract
We solve the following three problems. 1. How much can the radial growth of an entire function $f$ be reduced by multiplying it by some nonzero entire function? We give the answer in terms of the growth of the integral means of $\ln|f|$ over the circles centered at the origin. 2. We estimate the smallest possible radial growth of non zero entire functions that vanish on a given distribution of points $Z$. We solve this problem in terms of the growth of the radial integral counting function of $Z$. 3. Let $F=f/g$ be a meromorphic function with representations as the ratio of entire functions $f\neq 0$ and $g\neq 0$. How small can the radial growth of entire functions $f$ and $g$ be in such representations in relation to the growth of the Nevanlinna characteristic of $F$? All solutions have a non-asymptotic uniform character, and the obtained inequalities are sharp. All of them are based on some main theorem for subharmonic functions, which relies on the Govorov--Petrenko--Dahlberg--Essén inequality and uses our general results on the existence of subharmonic minorants.