This paper studies the unicity of meromorphic(resp. entire) functions of the form $f^n{f}^{\text{'}}$ and obtains the following main result: Let $f$ and $g$ be two non-constant meromorphic (resp. entire) functions, and let $a\in ℂ\setminus \left\{0\right\}$ be a non-zero finite value. Then, the condition that $E_{3)}(a,f^n{f}^{\text{'}})=E_{3)}(a,g^n{g}^{\text{'}})$ implies that either $f=dg$ for some $(n+1)$-th root of unity $d$, or $f=c_1{e}^{cz}$ and $g=c_2{e}^{-cz}$ for three non-zero constants $c$, $c_1$ and $c_2$ with ${\left(c_1{c}_2\right)}^{n+1}{c}^2=-a^2$ provided that $n\ge 11$ (resp. $n\ge 6$). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed. ...

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation $$\begin{array}{cc}& f^{\left(k\right)}+A_{k-1}{f}^{(k-1)}+\cdots +A_2{f}^{\text{'}\text{'}}+(D_1\left(z\right)+A_1\left(z\right){\mathrm{e}}^{az})f^{\text{'}}\hfill \\ & +(D_0\left(z\right)+A_0\left(z\right){\mathrm{e}}^{bz})f=F(k\ge 2),\hfill \end{array}$$ where $a$, $b$ are complex constants that satisfy $ab(a-b)\ne 0$ and $A_j\left(z\right)$ $(j=0,1,\cdots ,k-1)$, $D_j\left(z\right)$ $(j=0,1)$, $F\left(z\right)$ are entire functions with $max\{\rho \left(A_j\right)(j=0,1,\cdots ,k-1),\rho \left(D_j\right)$ $(j=0,1)\}<1$. We also investigate the relationship between small functions and the solutions of the above equation.

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...