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arXiv · 1807.08619

On the cardinality of unique range sets with weight one

Bikash Chakraborty·Sagar Chakraborty
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Abstract

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 $f\equiv g$.

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Bikash Chakraborty, Sagar Chakraborty. 2018-07-19. On the cardinality of unique range sets with weight one. https://doi.org/10.1007/s11253-020-01849-z

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