On the cardinality of unique range sets with weight one
UDC 517.9 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 $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \tex...
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| Date: | 2020 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2020
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6022 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.9
Two meromorphic functions $f$ and $g$ are said to share the set $S\subset \mathbb{C}\cup\{\infty\}$ with weight $l\in\mathbb{N}\cup\{0\}\cup\{\infty\},$ if $E_{f}(S,l)=E_{g}(S,l),$ where $$$E_{f}(S,l)=\bigcup_{a \in S} \big \{(z,t) \in \mathbb{C}\times\mathbb{N} \bigm| f(z)=a \; \text{with multiplicity} \;p \big \},$$ 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 [Indian J.  Pure and Appl.  Math., 31, No~4, 431–440 (2000)] by showing that there exist a finite set $S$ with 13 elements such that $E_{f}(S,1)=E_{g}(S,1)$ implies $f\equiv g.$ |
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| DOI: | 10.37863/umzh.v72i7.6022 |