On the Characterisations of a New Class of Strong Uniqueness Polynomials Generating Unique Range Sets

The purpose of the paper is to introduce a new class of strong uniqueness polynomials with the best possible answer to the question posed by the first author (Banerjee, Ann Acad Sci Fenn Math 40: 465-474, 2015). We find the corresponding unique range set which improves a result due to Frank-Reinders (Complex Var Theory Appl 37(1): 185-193, 1998). At the time of characterising the strong uniqueness polynomial, we also encounter the cases where the polynomial is not necessarily critically injective and find its degree and the corresponding unique range set. Furthermore, we find some connections between this new class of polynomials and most of the polynomials generating URSM introduced so far in the literature of value distribution theory. As an application of our results, we will show that the theorems improve many previous results. Lastly, we rectify the gap in the last section of An (Acta Math Vietnam 27(3), 251-256, 2002, p. 255, 1. 11-14) and thus improve the result obtained by An (Acta Math Vietnam 27(3), 251-256, 2002) significantly.