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arXiv · 2503.03296

Decrease in growth of entire and meromorphic functions

B. N. Khabibullin
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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.

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B. N. Khabibullin. 2025-03-05. Decrease in growth of entire and meromorphic functions. https://arxiv.org/abs/2503.03296

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