On the uniqueness problems of entire functions and their 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.

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 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.

Let f be a transcendental entire function, and let {U,V subset mathbb{C}} be disjoint simply-connected domains. Must one of {f^{-1}(U)} and {f^{-1}(V)} be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains. (A domain {Usubset mathbb{C}} is completely invariant under f if {f^{-1}(U)=U}.) It was recently observed by Julien Duval that Baker's argument, which has also been used in later generalisations and extensions of Baker's result, contains a flaw. We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function {f(z)= e^z+z}, there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property. Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f) is bounded; in other words, f belongs to the Eremenko–Lyubich class. On the other hand, if S(f) is finite (or if certain additional hypotheses are imposed), many of the original results do hold. For the convenience of the research community, we also include a description of the error in Baker's proof, and a summary of other papers that are affected.

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).

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].

W. Bergweiler proved in 1990 that the equation f∘g= Q has infinitely many solutions in the complex plane, provided f and g are transcendental entire functions and Q is a non-constant polynomial. We prove a related result, using standard notations from the Nevanlinna theory: Let f be a non-linear entire function of finite order of growth, let g be a transcendental entire function of finite lower order and let Q be a non-constant entire function such that T(r,Q)=S(r,g). If II denotes the canonical product formed with the zeros of f∘g Q, then T(r,φ) ≠ S(r,g). As an immediate consequencef∘g−Q has infinitely many zeros.

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

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.

A classic theorem of Polya shows that 2 z is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of Gel’fond concerns entire functions that are integer valued on the setX a={a n:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed.

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set J(f) are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions fK, K ∈ N, such that the box and packing dimensions of JfK are equal to two, even though as K → ∞ the Hausdorff dimension of JfK tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function. 2000 Mathematics Subject Classification 30D05, 37F10, 37F15, 37F35, 37F50.

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.

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.

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.