On the cardinality of unique range sets with weight one

B. Chakraborty; S. Chakraborty

doi:10.37863/umzh.v72i7.6022

B. Chakraborty Ramakrishna Mission Vivekananda Centenary College, West Bengal, India https://orcid.org/0000-0002-3951-4792
S. Chakraborty Jadavpur Univ., West Bengal, India

DOI: https://doi.org/10.37863/umzh.v72i7.6022

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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