A Non-zero Value Shared by an Entire Function and its Linear Differential Polynomials

An entire function sharing one nonzero value with its linear differential polynomials

Entire functions that share one value with their linear differential polynomials

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study a transcendental entire function f (z) that shares a non-zero polynomial a (z) with f′(z), together with its linear differential polynomials of the form: L[f] = a2(z)f″(z) + a3 (z)f′′′(z) + … + am (z)f(m) (z) (am (z) $\not\equiv$ 0), where the coefficients ak (z) (k = 2, 3, ..., m) are rational functions.

In this paper we investigate the uniqueness problem of entire function \(f\) and its linear differential polynomial\begin{equation*}a_{k}\left( z\right) f^{\left( k\right) }+a_{k-1}\left( z\right) f^{\left(k-1\right) }+\cdots+a_{1}\left( z\right) f'\end{equation*}sharing an entire function \(a\equiv a\left( z\right)\) counting multiplicities(CM) with\begin{equation*}\sigma \left( a\right) <\sigma \left( f\right)\end{equation*}under some restrictions imposed on the coefficients \(a_{j}\left( z\right) \left( {j=1,2,\dots,k}\right)\). Our result improves and generalizes some earlier results.

Using the results of S. S. Bhoosnurmath, we mainly study the uniqueness of entire and meromorphic functions that share small functions with their homogeneous and linear differential polynomials. In this paper, we obtain significant improvements and generalizations of the results of H. X. Yi.

We study the uniqueness question of transcendental meromorphic functions sharing three distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane, and study the uniqueness question of transcendental entire functions sharing two distinct finite values with their linear differential polynomials in some angular domains instead of the whole complex plane. The results in this paper improve the corresponding results from Frank and Weissenborn (Complex Var 7:33–43, 1986), Frank and Schick (Results Math 22:679–684, 1992), Bernstein et al. (Forum Math 8:379–396, 1996) and improve Theorem 3 in Zheng (Can Math Bull 47:152–160, 2004).

In this paper we consider an entire function when it shares a polynomial with its linear differential polynomial. Our result is an improvement of a result of P. Li.

We study the uniqueness of entire functions, when they share a linear polynomial, in particular, fixed points, with their linear differential polynomials.

In this paper, we study a uniqueness question of meromorphic functions concerning certain linear differential polynomials that share a nonzero finite value with the same of L-functions. The results in this paper extend the corresponding results from Li[6] and Li & Li[7].

In this paper, we study the growth of all solutions of a linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its linear differential polynomials having the same fixed points. The results in this paper also improve some known results. Two example are provided to show that the results in this paper are best possible.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by \begin{equation*} L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0. \end{equation*}In this paper, we solve an open problem posed by Li [J. Math. Anal. Appl. 310 (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials $ L_k[f] $. Furthermore, we study the Fermat-type functional equations of the form $ f^n+g^n=1 $ to find the meromorphic solutions (f, g) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.

In this paper, we study the growth of all solutions of a linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its linear differential polynomials having the same fixed points. The results in this paper also improve some known results. Two example are provided to show that the results in this paper are best possible.

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.

In this note, the relationship between a non-constant entire function $f$ and its linear differential polynomial $L(f)$ has been obtained when they share two finite values, ignoring multiplicities, by applying value distribution theory. This confirms Frank's conjecture as a special case. Entire solutions of certain types of non-linear differential equations are also discussed.

Uniqueness of Entire Functions that Share an Entire Function of Smaller Order with One of Their Linear Differential Polynomials