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The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The result of the paper improves and generalizes the recent results due to X. B. Zhang and J. F. Xu (2011). We also solve an open problem posed in the last section of X. B. Zhang and J. F. Xu (2011).

Using the notion of weighted sharing of values which was introduced by Lahiri (2001), we deal with the uniqueness problem for meromorphic functions when two certain types of nonlinear differential monomials namely $h^n{h}^{\left(k\right)}$ $(h=f,g)$ sharing a nonzero polynomial of degree less than or equal to $3$ with finite weight have common poles and obtain two results. The results in this paper significantly rectify, improve and generalize the results due to Cao and Zhang (2012).

This paper studies the uniqueness of meromorphic functions $$f^n\prod _{i=1}^k{\left(f^{\left(i\right)}\right)}^{n_i}\text{and}{g}^n\prod _{i=1}^k{\left(g^{\left(i\right)}\right)}^{n_i}$$ that share two values, where $n,n_k,k\in \mathbb{N}$, $n_i\in \mathbb{N}\cup \left\{0\right\}$, $i=1,2,...,k-1$. The results significantly rectify, improve and generalize the results due to Cao and Zhang (2012).

In the paper we discuss the uniqueness of the $n$-th power of a meromorphic function sharing a small function with the power of its $k$-th derivative and improve and supplement a result of Zhang-Lü [Complex Var. Elliptic Equ. 53 (2008), no. 9, 857–867]. We also rectify one recent result obtained by Chen and Zhang in [Kyungpook Math. J. 50 (2010), no. 1, 71–80] dealing with a question posed by T.D. Zhang and W.R. Lü in [Complex Var. Elliptic Equ. 53 (2008), no. 9, 857–867].

The motivation of this paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial with the help of the idea of normal family. The result of the paper improves and generalizes the recent result due to Zhang and Xu [24]. Our another remarkable aim is to solve an open problem as posed in the last section of [24].

The purpose of the paper is to study the uniqueness problems of linear differential polynomials of entire functions sharing a small function and obtain some results which improve and generalize the related results due to J. T. Li and P. Li (2015). Basically we pay our attention to the condition $\lambda \left(f\right)\ne 1$ in Theorems 1.3, 1.4 from J. T. Li and P. Li (2015). Some examples have been exhibited to show that conditions used in the paper are sharp.

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 establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P\left(f\right){\prod }_{i=1}^k{\left(f^{\left(i\right)}\right)}^{n_i}$ and $P\left(g\right){\prod }_{i=1}^k{\left(g^{\left(i\right)}\right)}^{n_i}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.

In connection to a conjecture of W. Lü, Q. Li and C. Yang (2014), we prove a result on small function sharing by a power of a meromorphic function with few poles with a derivative of the power. Our results improve a number of known results.

We investigate the uniqueness problem of entire functions that share two polynomials with their $k$th derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible.