Let f(z), $z=r{e}^{i\theta }$, be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values $I_{\delta }\left(r\right)$ and the iterated mean values $N_{\delta ,k}\left(r\right)$ of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^n\left(z\right)+P_d(z,f)=p_1\left(z\right){\mathrm{e}}^{{\alpha }_1\left(z\right)}+p_2\left(z\right){\mathrm{e}}^{{\alpha }_2\left(z\right)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f\left(z\right)$ of degree $d\le n-1$ with small functions of $f\left(z\right)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and ${\alpha }_1$, ${\alpha }_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ρ, then every asymptotic value of f, except at most 2ρ of them, is a limit point of critical values of f.We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f'fn with n ≥ 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration...

We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.

The purpose of this paper is to solve two functional equations for generalized Joukowski transformations and to give a geometric interpretation to one of them. Here the Joukowski transformation means the function $1/2(z+z^{-1})$ of a complex variable z.

We deal with a uniqueness theorem of two meromorphic functions that share three values with weights and also share a set consisting of two small meromorphic functions. Our results improve those by G. Brosch, I. Lahiri & P. Sahoo, T. C. Alzahary & H. X. Yi, P. Li & C. C. Yang, and others.

Dans cet article, nous démontrons deux résultats. L’un concerne les séries $f^{\text{'}}\left(z\right)=\sum a\left(n\right)z^n/n!$ telles que $\sum a\left(n\right)x^n$ est une série algébrique. Soit $AE$ cet ensemble de fonctions. Si $f$ appartient à $AE$, et si $g\left(z\right)$ est un polynôme-exponentiel tel que $h\left(z\right)=f\left(z\right)/g\left(z\right)$ est entière, alors il existe un polynôme $P\left(z\right)$ tel que $P\left(z\right)h\left(z\right)$ appartienne à $AE$.L’autre résultat est parallèle au premier. Soit $\sum u\left(n\right)x^n$ une série algébrique à coefficients dans un corps $𝕂$ (qui est soit $𝕂$, soit un corps quadratique imaginaire). Soit $\sum v\left(n\right)x^n$ une série rationnelle à coefficients dans $𝕂$. Avec...