On the cardinality of unique range sets with weight one

Abstract

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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Published
15.07.2020
How to Cite
Chakraborty, B., and S. Chakraborty. “On the Cardinality of Unique Range Sets With Weight One”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 7, July 2020, pp. 997-1005, doi:10.37863/umzh.v72i7.6022.
Section
Research articles