Abstract

In the paper we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain a new result regarding the structure and position of uniqueness. This new result has immense application like classifying different expressions to be or not to be unique. The principle objective of the paper is to study the uniqueness of meromorphic functions when sharing a value with weight $l(\geq 0)$ and its nonlinear differential polynomials. We prove a result which significantly generalize the result of Waghamore and Maligi [Commun. Math. {\bf 28} (2020), 289-299] by considering the difference polynomials of the form $f^n(f -1)^mP[f]$ and citing two proper examples we have shown that the result is true only for a particular case. Finally we present the compact version of the same result as an improvement.

Keywords and Phrases

Uniqueness, Entire function, difference-differential polynomial.

A.M.S. subject classification

Primary 30D35, Secondary 39A10

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