Abstract
The purpose of the paper is to introduce a new class of strong uniqueness polynomial satisfying Fujimoto's conditions. We shall also show that for a suitably prescribing value of the constant term in the class of polynomials, an SUPM (SUPE) with smallest degree ever can be obtained. We also find corresponding class of unique range sets generated from those of polynomials under weaker sharing hypothesis. Finally, rectifying the results in the application part of (2) we shall reduce the cardinalities of the range sets under some assumptions on the deficiencies of the functions. At the last section we shall place an open question for future research.
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