Transcendental Meromorphic Function Research Articles

A result on a question of Lü, Li and Yang

Let f be a transcendental meromorphic function of finite order with finitely many poles, c ? C\{0} and n,k ? N. Suppose fn(z)-Q1(z) and (fn(z+c))(k)- Q2(z) share (0,1) and f(z), f(z+c) share 0 CM. If n ? k + 1, then (fn(z+c))(k) ? Q2(z)/Q1(z)fn(z), where Q1, Q2 are polynomials with Q1Q2 ?/ 0. Furthermore, if Q1 = Q2, then f(z)=c1e?/n z, where c1 and ? are non-zero constants such that e?c = 1 and ?k = 1. Also we exhibit some examples to show that the conditions of our result are the best possible.

Herman rings with small periods and omitted values

All possible arrangements of cycles of three periodic as well as four periodic Herman rings of transcendental meromorphic functions having at least one omitted value are determined. It is shown that if p = 3 or 4, then the number of p-cycles of Herman rings is at most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycle of Herman rings simultaneously. Finally some examples of functions having no Herman ring are discussed.

Uniqueness of Meromorphic Functions Sharing Values with their nth Order Exact Differences

Let f(z) be a transcendental meromorphic function in the complex plane of hyper-order strictly less than 1. It is shown that if f(z) and its nth exact difference Δnf(z) (≢ 0) share three distinct periodic functions $${\rm{a, b, c}} \in \mathcal{\hat{S}}(f)$$ with period 1 CM, where $$\mathcal{\hat{S}}(f) = \mathcal{S}(f)\cup\{{\infty}\}$$ and $$\mathcal{S}(f)$$ denotes the set of all small functions of f(z), then Δnf(z) ≡ f(z).

Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

For a transcendental meromorphic function f ( z ) , the main aim of this paper is to investigate the properties on the zeros and deficiencies of some differential-difference polynomials. Some results about the deficiencies of some differential-difference polynomials concerning Nevanlinna deficiency and Valiron deficiency are obtained, which are a generalization of and improvement on previous theorems given by Liu, Lan and Zheng, etc.

Simply connected Baker domains of some meromorphic functions

The first example of a meromorphic function with non-invariant Baker domains was published by Baker et al. (1991) [2]. In this article we answer some questions concerning connectivity of Baker domains of that meromorphic function and location of the iterates of critical values. We analyze dynamics on the Fatou set and connectivity of the Julia set. We extend some of the achieved results to the family fλ,μ(z)=λexp(z)+μz of transcendental meromorphic functions, where λ,μ∈C∖{0}. We also prove that some ‘non-parabolic’ bifurcations appear many times in this family.

On the iteration of quasimeromorphic mappings

AbstractThe Fatou–Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.

Hausdorff measure of escaping sets on certain meromorphic functions

We consider transcendental meromorphic function for which the set of finite singularities of its inverse is bounded. Bergweiler and Kotus gave bounds for the Hausdorff dimension of the escaping sets if the function has no logarithmic singularities over ∞, the multiplicities of poles are bounded and the order is finite. We study the case of infinite order and find gauge functions for which the Hausdorff measure of escaping sets is zero or ∞.

Families of Baker domains for meromorphic functions with countable many essential singularities

Given a transcendental meromorphic function that satisfies an asymptotic behaviour near infinity, Rippon and Stallard showed that such function has a family of invariant Baker domains associated to its essential singularity at infinity. In this work, we extend this result to meromorphic functions with a countable number of essential singularities. Furthermore, for any we provide a closed form description of a large class of functions with countable many essential singularities that exhibit p families of invariant Baker domains. This closed form allow us to provide explicit examples of the results.

Normal family of meromorphic functions sharing holomorphic functions and the converse of the bloch principle

In 1996, C. C. Yang and P. C. Hu [8] showed that: Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n ≥ 2, n1, …,nk are nonnegative integers such that n1+⋯+nk≥1; thus fn(f′)n1⋯(f(k))nk−ahas infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that fn(f′)n1⋯(f(k))nk−a(z) has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ≡ 0 is a small function of f and n ≥ 2, n1,…,nk are nonnegative integers satisfying n1+…+ nk ≥ 1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.

Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function

In this paper, we study the value distribution of differential polynomial with the form \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)},\) where f is a transcendental meromorphic function. Namely, we prove that \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)}-P(z)\) has infinitely zeros, where P(z) is a nonconstant polynomial and \(n\in {\mathbb {N}},\)\(k, n_1, \dots , n_k, t_1, \dots , t_k\) are positive integer numbers satisfying \(n+\sum _{v}^{k}n_v\ge \sum _{v=1}^{k}t_v+3.\) Using it, we establish some normality criterias for family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. Our results generalize some previous results on normal family of meromorphic functions.

Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

AbstractIn this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

Chaos in dynamics of a family of transcendental meromorphic functions

In this paper, the chaotic behaviours in the real and complex dynamics of $\zeta_\lambda(z)=\lambda\frac{z}{z+1} e^{-z}$, $\lambda >0, z\in \mathbb{C}$ are investigated. The bifurcation in the dynamics of $\zeta_\lambda(x), x\in {\mathbb{R}}\setminus \{-1\}$, occurs at several parameter values and the dynamics becomes chaotic when the parameter $\lambda$ crosses certain values. The Lyapunov exponent of $\zeta_\lambda(x)$ is computed for quantifying the chaos. The characterization of the Julia set of $\zeta_\lambda(z)$ as complement of the basin of attraction is found and is applied to computationally simulate the images of the Julia sets. Finally, the results on the dynamics of $\zeta_\lambda(z)$ are compared with the known results.

The Schwarzian derivative and the Wiman-Valiron property

Consider a transcendental meromorphic function in the plane with finitely many critical values, such that the multiple points have bounded multiplicities and the inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that if the Schwarzian derivative is transcendental then the function has infinitely many multiple points, the inverse function does not have a direct transcendental singularity over infinity, and infinity is not a Borel exceptional value. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.

Notes on the value distribution of ff(k)-b

Let f denote a transcendental meromorphic function with N(r, f) = S(r, f) and k be an integer. By using methods different from others, we have been able to derive several new results and pose some new conjectures that relate to the yet to be resolved conjecture concerning the quantitative estimates on the zeros of ff(k)-b, for a non-vanishing small function b.

On permutable meromorphic functions

We study the class $\mathcal{M}$ of functions meromorphic outside a countable closed set of essential singularities. We show that if a function in $\mathcal{M}$, with at least one essential singularity, permutes with a non-constant rational map $g$, then $g$ is a M\"{o}bius map that is not conjugate to an irrational rotation. For a given function $ f \in\mathcal{M}$ which is not a M\"{o}bius map, we show that the set of functions in $\mathcal{M}$ that permute with $ f $ is countably infinite. Finally, we show that there exist transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C}$ such that, among functions meromorphic in the plane, $f$ permutes only with itself and with the identity map.

A note on the value distribution of $f^1(f^{(k)})^n$

Let $f$ be a transcendental meromorphic function in the complex plane $\mathbf{C}$, and a be a nonzero complex number. We give quantitative estimates for the characteristic function $T(r,f)$ in terms of $N(r,1/( f^1(f^{(k)})^n-a))$, for integers $k$, $l$, $n$ greater than 1. We conclude that $f^1(f^{(k)})^n$ assumes every nonzero finite value infinitely often.

Derivatives of meromorphic functions and polynomial functions

Let be a polynomial, and let f be a transcendental meromorphic function. If all but finitely many zeros of f have multiplicity at least k, and there exists such that whenever , then has infinitely many zeros in the complex plane.

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

Herman rings of meromorphic maps with an omitted value

We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles one of which is an omitted value have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.

Value Distribution of the Product of a Meromorphic Derivative and a Power of the Function

In the paper we discuss the value distribution of the product of the derivative of a transcendental meromorphic function and a power of the function.