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.

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].