Abstract
A finite set S in the extended complex plane is called a unique range set counting multiplicities (URSCM) (unique range set ignoring multiplicities, URSIM) for meromorphic functions on if any two nonconstant meromorphic functions f and g in satisfying counting multiplicities (CM) (ignoring multiplicities IM), then f = g. The question ‘What is the smallest cardinality for such a set?’ is still a challenging and open one. In this article, for meromorphic functions on we exhibit a URSCM and a URSIM of 13 and 19 elements, respectively. We also characterize a unique range set for entire functions of order <1 on . Some conjectures are proposed for further studies. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.
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