Decrease in growth of entire and meromorphic functions
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arXiv
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| Format: | Preprint |
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2025
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| author | Khabibullin, B. N. |
| author_facet | Khabibullin, B. N. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decrease in growth of entire and meromorphic functions Khabibullin, B. N. Complex Variables 30D35 (Primary) 30D20, 30D30, 31A05 (Secondary) 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. |
| title | Decrease in growth of entire and meromorphic functions |
| topic | Complex Variables 30D35 (Primary) 30D20, 30D30, 31A05 (Secondary) |
| url | https://arxiv.org/abs/2503.03296 |