Uniqueness of $P(f)$ and $[P(f)]^{(k)}$ concerning weakly weighted sharing

TL;DR

This paper investigates the uniqueness of polynomial expressions of meromorphic functions sharing small functions, improving previous results and identifying conditions for the polynomial to be a monomial, with sharp inequalities and examples.

Contribution

It introduces new conditions under which polynomial expressions of meromorphic functions are uniquely determined, extending prior work with sharper results.

Findings

Conditions for $P(f)$ to be a non-zero monomial

Specific form of $f$ under these conditions

Sharp inequalities demonstrated with examples

Abstract

In this paper, with the help of the idea of weakly weighted sharing introduced by \emph{Lin -Lin} [Kodai Math. J., 29(2006), 269-280], we study the uniqueness of a polynomial expression $ P(f) $ and $ [P(f)]^{(k)} $ of a meromorphic function $ f $ sharing a small function. The main results significantly improved the result of \emph{Liu - Gu} [Kodai Math. J., 27(3)(2004), 272-279]. This research work explores certain condition under which the polynomial $ P(f) $ can be reduced to a non-zero monomial, and as a consequence, the specific form of the function $ f $ is obtained. By some constructive examples it has been shown that some conditions in the main results can not be removed and some of the inequalities are sharp.