On the cardinality of unique range sets with weight one
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866910679446847488 |
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| author | Chakraborty, Bikash Chakraborty, Sagar |
| author_facet | Chakraborty, Bikash Chakraborty, Sagar |
| contents | Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\}$, if $E_{f}(S,l)=E_{g}(S,l)$ where $$E_{f}(S,l)=\bigcup\limits_{a \in S}\{(z,t) \in \mathbb{C}\times\mathbb{N}~ |~ f(z)=a ~\text{with~ multiplicity}~ p\},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l$.
In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set $S$ with cardinality $\geq 13$ such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_08619 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On the cardinality of unique range sets with weight one Chakraborty, Bikash Chakraborty, Sagar Complex Variables 30D35 Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\}$, if $E_{f}(S,l)=E_{g}(S,l)$ where $$E_{f}(S,l)=\bigcup\limits_{a \in S}\{(z,t) \in \mathbb{C}\times\mathbb{N}~ |~ f(z)=a ~\text{with~ multiplicity}~ p\},$$ where $t=p$ if $p\leq l$ and $t=p+1$ if $p>l$. In this paper, we improve and supplement the result of L. W. Liao and C. C. Yang (On the cardinality of the unique range sets for meromorphic and entire functions, Indian J. Pure appl. Math., 31 (2000), no. 4, 431-440) by showing that there exist a finite set $S$ with cardinality $\geq 13$ such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g$. |
| title | On the cardinality of unique range sets with weight one |
| topic | Complex Variables 30D35 |
| url | https://arxiv.org/abs/1807.08619 |