In this paper, we investigate the growth of solutions of a certain class of linear differential equation where the coefficients are analytic functions in the closed complex plane except at a finite singular point. For that, we will use the value distribution theory of meromorphic functions developed by Rolf Nevanlinna with adapted definitions.

Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$$(i=1,......$

This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_1$, $f_2$, $f_3$ on an annulus $𝔸\left(R_0\right)$ share four distinct values regardless of multiplicity and have the complete identity set of positive counting function, then $f_1=f_2$ or $f_2=f_3$ or $f_3=f_1$. This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and...

Let f(z) be a finite order transcendental meromorphic function such that λ(1/f(z)) < σ(f(z)), and let c ∈ ℂ∖0 be a constant such that f(z+c) ≢ f(z) + c. We mainly prove that $max\tau \left(f\left(z\right)\right),\tau \left({\Delta }_{c}f\left(z\right)\right)=max\tau \left(f\right(z\left)\right),\tau \left(f\right(z+c\left)\right)=max\tau \left({\Delta }_{c}f\left(z\right)\right),\tau \left(f(z+c)\right)=\sigma \left(f\left(z\right)\right)$, where τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).

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).

In the paper based on the question of Zhang and Lü [15], we present one theorem which will improve and extend results of Banerjee-Majumder [2] and a recent result of Li-Huang [9].

The motivation of this paper is to study the uniqueness problems of meromorphic functions concerning differential polynomials that share a small function. The results of the paper improve and generalize the recent results due to Fengrong Zhang and Linlin Wu [13]. We also solve an open problem as posed in the last section of [13].

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.

We investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Our results extend the previous results due to Peng and Chen.