Soit $q$ dans $\mathbb{Z}$ tel que $\left|q\right|\ge 2$. Dans cette note, nous démontrons que si une fonction entière $f$ a une croissance assez lente et si $f(q^n+i{q}^m)\in \mathbb{Z}\left[i\right]$ pour $n,m\in \mathbb{N}$, alors $f$ est un polynôme.

We deal with the problem of uniqueness of meromorphic functions sharing three values, and obtain several results which improve and extend some theorems of M. Ozawa, H. Ueda, H. X. Yi and other authors. We provide examples to show that results are sharp.

Let $k$ be a nonnegative integer or infinity. For $a\in ℂ\cup \left\{\infty \right\}$ we denote by $E_k(a;f)$ the set of all $a$-points of $f$ where an $a$-point of multiplicity $m$ is counted $m$ times if $m\le k$ and $k+1$ times if $m>k$. If $E_k(a;f)=E_k(a;g)$ then we say that $f$ and $g$ share the value $a$ with weight $k$. Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to Xia and Xu (2011)...

The purpose of the paper is to represent the two shared set problems in an elaborative and convenient manner. In the main result of the paper, we have exhaustively treated the two shared set problem on the open complex plane. As a consequence of the main result, we have investigated the same problem in a different perspective, which has yet not been studied. Further, two examples have been exhibited in the paper to show the sharpness of some of these results.

We study the uniqueness of meromorphic functions using nonlinear differential polynomials and the weighted value sharing method. Though the main concern of the paper is to improve a recent result of L. Liu [Comput. Math. Appl. 56 (2008), 3236-3245], as a consequence of the main result we also improve and generalize some former results of T. Zhang and W. Lu [Comput. Math. Appl. 55 (2008), 2981-2992], A. Banerjee [Int. J. Pure Appl. Math. 48 (2008), 41-56] and a recent result of the present author...

Let f and g be entire functions, n, k and m be positive integers, and λ, μ be complex numbers with |λ| + |μ| ≠ 0. We prove that ${\left(fⁿ\left(z\right)(\lambda f^m\left(z\right)+\mu )\right)}^{(}k)$ must have infinitely many fixed points if n ≥ k + 2; furthermore, if ${\left(fⁿ\left(z\right)(\lambda f^m\left(z\right)+\mu )\right)}^{(}k)$ and ${\left(gⁿ\left(z\right)(\lambda g^m\left(z\right)+\mu )\right)}^{(}k)$ have the same fixed points with the same multiplicities, then either f ≡ cg for a constant c, or f and g assume certain forms provided that n > 2k + m* + 4, where m* is an integer that depends only on λ.

We study the uniqueness of entire functions which share a value or a function with their first and second derivatives.

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 results....