On the cardinality of unique range sets with weight one

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 $$E_f(S,l)=\bigcup _{a\in S}\{(z,t)\in \mathbb{C}\times \mathbb{N}|f(z)=a\text{with multiplicity}p\},$$ where $t=p$ if $p\le l$ 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 $\ge 13$ such that $E_f(S,1)=E_g(S,1)$ implies $f\equiv g$.

10 pages. arXiv admin note: text overlap with arXiv:1711.08808, arXiv:1608.02125