In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f\left(z\right)$ and $g\left(z\right)$ be two transcendental entire functions of finite order, and $\alpha \left(z\right)$ a small function with respect to both $f\left(z\right)$ and $g\left(z\right)$. Suppose that $c$ is a non-zero complex constant and $n\ge 7$ (or $n\ge 10$) is an integer. If $f^n\left(z\right)(f\left(z\right)-1)f(z+c)$ and $g^n\left(z\right)(g\left(z\right)-1)g(z+c)$ share “$\left(\alpha \right(z),2)$” (or ${(\alpha \left(z\right),2)}^{*}$), then $f\left(z\right)\equiv g\left(z\right)$. Our results extend and generalize some well known previous results....

This paper deals with the uniqueness problem for meromorphic functions sharing one value with finite weight. Our results generalize those of Fang, Hong, Bhoosnurmath and Dyavanal.

With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^n{\left(f^{\left(k\right)}\right)}^m-p$ and $g^n{\left(g^{\left(k\right)}\right)}^m-p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018).

We prove some uniqueness theorems for meromorphic functions and their derivatives that share a meromorphic function whose order is less than those of the above meromorphic functions. The results in this paper improve those given by G. G. Gundersen & L. Z. Yang, J. P. Wang, J. M. Chang & Y. Z. Zhu, and others. Some examples are provided to show that our results are the best possible.

We prove a result on the uniqueness of meromorphic functions sharing three values with weights and as a consequence of this result we improve a recent result of W. R. Lü and H. X. Yi.

We prove a uniqueness theorem for meromorphic functions involving linear differential polynomials generated by them. As consequences of the main result we improve some previous results.

In the paper we deal with the uniqueness of meromorphic functions when two non-linear differential polynomials generated by two meromorphic functions share a small function.

We investigate the uniqueness of a $q$-shift difference polynomial of meromorphic functions sharing a small function which extend the results of N. V. Thin (2017) to $q$-difference operators.

We deal with the uniqueness problem for analytic functions sharing four distinct values in an angular domain and obtain some theorems which improve the result given by Cao and Yi [J. Math. Anal. Appl. 358 (2009)].

We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${\left(Q\left(f\right)\right)}^{\left(k\right)}$ and ${\left(Q\left(g\right)\right)}^{\left(k\right)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{\text{'}}$. The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).