On the distribution of unique range sets and its elements over the extended complex plane

Abstract

In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular,identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets.