An Entire Function that Shares a Small Function with its Derivative and Linear Differential Polynomial

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.

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

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.

We study the relationship between an entire function f and its certain type of linear differential polynomial L when f and L share one finite nonzero value under some additional conditions. The results improve and generalize some previous results obtained by C.C. Yang and some other authors.

In this paper, we investigate meromorphic functions that share a small function with one of its linear differential polynomials and prove several theorems which generalize and improve the main results given by J. L. Zhang and L. Z. Yang. Some examples are provided to show that the results in this paper are sharp.

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].

We use the idea of normal family to investigate the uniqueness problems for entire functions: $f$ shares one finite non-zero value with its first derivative $f^{(1)}$ together with its certain type of linear differential polynomial $L(f)$ of constant coefficients. The results here improve and generalize results obtained by Yang, Li and Yang, and Li

This paper is devoted to studying the uniqueness problem of entire functions sharing a small function with their linear differential polynomials.

Linear complete differential resultants and the implicitization of linear DPPEs

Linear sparse differential resultant formulas

We apply Rossi’s half-plane version of Borel’s theorem to study the zero distribution of linear combinations of A \mathcal {A} -entire functions (Theorem 1.2). This provides a unified way to study linear q q -difference, difference and differential operators (with entire coefficients) preserving subsets of A \mathcal {A} -entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite ( q q -)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of A \mathcal {A} -entire functions.

This paper is devoted to studying the growth and the oscillation of solutions\nof the second order non-homogeneous linear differential equation\n$$\nf''+A_{1} (z) e^{P (z)}f'+A_{0} (z) e^{Q (z)}f = F,\n$$\nwhere $P (z)$, $Q (z)$ are nonconstant polynomials such that $\\deg P=\\deg Q=n$ and $A_{j} (z)$ $( \\not\\equiv 0 )$ $(j=0,1)$,\n$F\\not\\equiv 0$ are entire functions with $\\rho ( A_{j} ) < n$ $( j=0,1 )$.\nWe also investigate the relationship between small functions and differential polynomials\n$g_{f} (z)=d_{2}f''+d_{1}f'+d_{0}f$, where $d_{0} (z)$, $d_{1} (z)$, $d_{2} (z)$ are entire functions\nthat are not all equal to zero with $\\rho ( d_{j} ) < n$ $( j=0,1,2 )$ generated by solutions of the above equation.

Given an entire function f of finite order , let L(z, f ) = m j=0 b j (z) f (k j ) (z+c j ) be a linear delay-differential polynomial of f with small coefficients in the sense of O(r + ) + S(r, f ), < .Provided and are similar small functions, we consider the zero distribution of L(z, f ) - f n for n 3 and n = 2, respectively.Our results are improvements and complements of Chen [

We give a slightly different proof of a theorem by N, Steinmetz: Suppose that f is a transcendental meromorphic function Then for ∊ > 0 and distinct functions a1...aq such that T(r. aj) = 0(T(r,f) we have For the proof we use a lemma on the zeros of linear differential polynomials of the form W(a1…aq,f).

Entire or meromorphic functions sharing one value