By using an extension of the spherical derivative introduced by Lappan, we obtain some results on normal functions and normal families, which extend Lappan's five-point theorems and Marty's criterion, and improve some previous results due to Li and Xie, and the author. Also, another proof of Lappan's theorem is given.

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).

We investigate how the growth of an algebroid function could be affected by the distribution of the arguments of its a-points in the complex plane. We give estimates of the growth order of an algebroid function with radially distributed values, which are counterparts of results for meromorphic functions with radially distributed values.

In the paper we consider the growth of entire solution of a nonlinear differential equation and improve some existing results.

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...$