Meromorphic functions bi-weighted weakly sharing pairs of small functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866916049884020736 |
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| author | Quang, Si Duc Anh, Phung Nguyen Ngoc |
| author_facet | Quang, Si Duc Anh, Phung Nguyen Ngoc |
| contents | Two meromorphic functions $f$ and $g$ are said to weakly share a small function $a$ with bi-weight $(n,k)$ if the functions $f-a$ and $g-a$ have the same zeros with multiplicities truncated at level $n+1$, while zeros whose multiplicities exceed $k$ are disregarded. In this article, we show that if $f$ and $g$ weakly share three distinct small functions with suitable bi-weights and are not related by a quasi-Möbius transformation, then for every other small function $c$, the counting function $N(r,ν_f^c)$ is asymptotically equivalent to the characteristic function $T(r,f)$. Moreover, the truncated counting function $N_{(3}(r,ν_f^c)$, which counts only zeros of multiplicity at least $3$, is negligible. As an application, we further prove that $f$ and $g$ must be related by a quasi-Möbius transformation provided that they satisfy an additional condition, which is weaker than the usual assumption that they share a fourth pair of small functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26987 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Meromorphic functions bi-weighted weakly sharing pairs of small functions Quang, Si Duc Anh, Phung Nguyen Ngoc Complex Variables Two meromorphic functions $f$ and $g$ are said to weakly share a small function $a$ with bi-weight $(n,k)$ if the functions $f-a$ and $g-a$ have the same zeros with multiplicities truncated at level $n+1$, while zeros whose multiplicities exceed $k$ are disregarded. In this article, we show that if $f$ and $g$ weakly share three distinct small functions with suitable bi-weights and are not related by a quasi-Möbius transformation, then for every other small function $c$, the counting function $N(r,ν_f^c)$ is asymptotically equivalent to the characteristic function $T(r,f)$. Moreover, the truncated counting function $N_{(3}(r,ν_f^c)$, which counts only zeros of multiplicity at least $3$, is negligible. As an application, we further prove that $f$ and $g$ must be related by a quasi-Möbius transformation provided that they satisfy an additional condition, which is weaker than the usual assumption that they share a fourth pair of small functions. |
| title | Meromorphic functions bi-weighted weakly sharing pairs of small functions |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2605.26987 |