Uniqueness of Meromorphic Functions With Nonlinear Differential Polynomials Sharing a Small Function IM

In the paper, we discuss the distribution of uniqueness and its elements over the extended complex plane from different polynomials of view. We obtain some new results regarding the structure and position of uniqueness. These new results have immense applications like classifying different expressions to be or not to be unique. The principal objective of the paper is to study the uniqueness of meromorphic functions when sharing a small function $a(z)$ IM with restricted finite order and its nonlinear differential polynomials. The lemma on the logarithmic derivative by Halburb and Korhonen (Journal of Mathematical Analysis and Applications, \textbf{314} (2006), 477--87) is the starting point of this kind of research. In this direction, the current focus in this field involves exploring unique results for the differential-difference polynomials of meromorphic functions, covering both derivatives and differences. Liu et al. (Applied Mathematics A Journal of Chinese Universities, \textbf{27} (2012), 94--104) have notably contributed to this research. Their research establishes that when $n \leq k + 2$ for a finite-order transcendental entire function $f$ the differential-difference polynomial$[f^{n}f(z+c)]^{(k)} - \alpha(z)$ has infinitely many zeros. Here, $\alpha(z)$ is characterized by its smallness relatively to $f$. Additionally, for two distinct meromorphic functions $f$ and $g$, both of finite order, if the differential-difference polynomials $[f^{n}f(z+c)]^{(k)}$\ and\ $[g^{n}g(z+c)]^{(k)}$ share the value $1$ in the same set, then $f(z)=c_1e^{dz},$ $g(z)=c_2e^{-dz}.$ We prove two results, which significantly generalize the results of Dyavanal and Mathai (Ukrainian Math. J., \textbf{71} (2019), 1032--1042), and Zhang and Xu (Comput. Math. Appl., \textbf{61} (2011), 722-730) and citing a proper example 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.