On an open problem of value sharing and differential equations

The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and prove the following result: Let $f$ be a nonconstant meromorphic function and $L$ a nonconstant linear differential polynomial generated by $f$. Suppose that $a = a(z)$ ($\not \equiv 0, \infty $) is a small function of $f$. If $f-a$ and $L-a$ share $0$ CM and \[ (k+1)\overline N(r, \infty ; f)+ \overline N(r, 0; f')+ N_{k}(r, 0; f')< \lambda T(r, f')+ S(r, f') \] for some real constant $\lambda \in (0, 1)$, then $ f-a=(1+ {c}/{a})(L-a)$, where $c$ is a constant and $1+{c}/{a} \not \equiv 0$.

Mues–Reinders proved that if a non-constant meromorphic function f shares three distinct finite values IM with its Kth (2≤ k≤ 50) linear differential polynomial L(f), then f≡L(f). Frank and Hua proved the same result under a weak condition k≥2. In the present paper, we will prove this result in the case that k = 1.

In this paper, we shall study the unicity of meromorphic functions definedover non-Archimedean fields of characteristic zero such that their valence func-tions of poles grow slower than their characteristic functions. If f is such afunction, and f and a linear differential polynomial P(f) of f, whose coeffi-cients are meromorphic functions growing slower than f, share one finite valuea CM, and share another finite value b (6= a) IM, then P(f) = f. 1 Introduction. In 1929, R. Nevanlinna studied the unicity of meromorphic functions in C. Thefive value theorem due to R. Nevanlinna states that if two non-constant meromor-phic functions f and gin C share five distinct complex numbers a j IM (ignoringmultiplicity), which meansf −1 (a j ) = g −1 (a j ), j= 1,2,...,5in the sense of sets, then it follows that f = g. The four value theorem of R.Nevanlinna states that if two non-constant meromorphic functions f and gin Cshare four distinct complex numbers a j CM (counting multiplicity), which meansf

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.

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.

Entire functions that share one value with their linear differential polynomials

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.

Abstract.In this paper we study uniqueness of entire functions sharing a non-zero finite value with linear differential polynomials and address a result of W.Wang and P. Li.

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

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

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 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.

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.

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) &lt;\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.

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.