Unique range sets for powers of meromorphic functions

S. Mallick1, D. Sarkar2
Department of Mathematics, Cooch Behar Panchanan Barma University, West Bengal, India
The prime concern of the paper is to deal with the notion of the unique range set for powers of meromorphic functions. As a consequence, we show that the lower bound of URSM (URSE) and URSM-IM (URSE-IM) can be significantly reduced up to cardinality $4$ $(4)$ from $11$ $(7)$ and $17$ $(10)$ respectively, for a class of power of meromorphic (entire) functions. Various applications of our main result also improve and generalize different results of Khoai-An-Lai (Internat. J. Math., 2018), Yi (Nagoya Math. J., 1995) and (J. Shandong Univ. Nat. Sci., 1998). Moreover, on the basis of some new notions introduced in the paper, our main result and its applications, we have partially reduced the known Gross Problem to a more narrow formulation and posed a number of open problems in the last section to unveil the least cardinality problem of unique range sets.
meromorphic function; entire function; uniqueness; unique range set; reduced unique range set
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