Uniqueness of meromorphic functions whose derivatives share four small functions

In this paper, we will prove some uniqueness theorems of meromorphic functions whose derivatives share four distinct small functions. The results in this paper improve those given by R. Nevanlinna, L. Yang, G.D. Qiu, and other authors. An example is provided to show that the results in this paper are best possible.     

Uniqueness results of meromorphic functions whose derivatives share four small functions

We prove an oscillation theorem of two meromorphic functions whose derivatives share four values IM. From this we obtain some uniqueness theorems, which improve the corresponding results given by Yang [16] and Qiu [10], and supplement results given by Nevanlinna [9] and Gundersen [3, 4]. Some examples are provided to show that the results in this paper are best possible.     

Normality and fixed-points of meromorphic functions

LetF be families of meromorphic functions in a domainD, and letR be a rational function whose degree is at least 3. If, for anyf∈ F, the composite functionR(f) has no fixed-point inD, thenF is normal inD. The number 3 is best possible. A new and much simplified proof of a result of Pang and Zalcman concerning normality and, shared values is also given.     

Normality and shared sets of meromorphic functions

The purpose of this paper is to investigate the normal families and shared sets of meromorphic functions. The results obtained complement the related results due to Fang, Liu and Pang.     

Normality and quasinormality of zero-free meromorphic functions

Let k, K ∈ N and F be a family of zero-free meromorphic functions in a domain D such that for each f ∈ F , f(k)-1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most ν = K k+1 , where ν is equal to the largest integer not exceeding K/k+1 . In particular, if K = k, then F is normal. The results are sharp.     

On the family of meromorphic functions whose derivatives omit a holomorphic function

In this paper,we continue to study the normality of a family of meromorphic functions without simple zeros and simple poles such that their derivatives omit a given holomorphic function.Such a family in general is not normal at the zeros of the omitted function.Our main result is the characterization of the non-normal sequences,and hence some known results are its corollaries.     

Normality criteria of meromorphic functions sharing one value

Let k be a positive integer and F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k+1. If for each pair (f, g) in F, ff(k) and gg(k) share a non-zero complex number a ignoring multiplicity, then F is normal in D.     

Normality criteria of meromorphic functions with multiple zeros

Take positive integers n,k?2. Let F be a family of meromorphic functions in a domain D⊂C such that each f∈F has only zeros of multiplicity at least k. If, for each pair (f,g) in F, fn(f(k)) and gn(g(k)) share a non-zero complex number a ignoring multiplicity, then F is normal in D.     

Normality and curvilinear limits of meromorphic functions and their applications

The paper contains two parts. In the first part, the behavior of meromorphic functions along arbitrary Jordan curves ending at a single boundary point is studied. The second part describes applications of the results of the first part to the study of the value distribution of meromorphic functions in terms of P-sequences.     

Normality of meromorphic functions with multiple zeros and shared values

In this paper, we study the normality of a family of meromorphic functions and general criteria for normality of families of meromorphic functions with multiple zeros concerning shared values are obtained.     

Non-real zeros of derivatives of meromorphic functions

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.     

Unicity of meromorphic functions and their derivatives

In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f^(k) (k>1), then f=f^(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.     

Julia directions of meromorphic functions and their derivatives

We prove that if a transcendental meromorphic function has no Julia direction and is bounded on a path to ￥ \infty then there is a common Julia direction for all derivatives. Related statements are obtained under the assumption that f is o(?{ | z | }) o(\sqrt{\mid z \mid}) or O(?{ | z | }) O(\sqrt{\mid z \mid}) on a path to ￥ \infty . Further we disprove a conjecture of Frank and Wang by means of a counterexample.     

Value sharing of meromorphic functions and their derivatives

In this paper, we estimate the size of ρn's in the famous L. Zalcman's lemma. With it, we obtain some uniqueness theorems for meromorphic functions f and f^′ when they share two transcendental meromorphic functions.     

Uniqueness and set sharing of derivatives of meromorphic functions

We prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share two or three sets which will improve some existing results.     

Normality and boundary behaviour of arbitrary and meromorphic functions along simple curves and applications

In this paper, we establish some theorems giving necessary and sufficient conditions for an arbitrary function defined in the unit disc of the complex plane to have boundary values along classes of an equivalence relation over simple curves. Our results generalize the well-known theorems on asymptotic and angular boundary behaviours of meromorphic functions (Lindelölf-, Lehto–Virtanen- and Seidel–Walsh-type theorems). The obtained results are applied to the study of boundary behaviour of meromorphic functions along curves using P-sequences, as well as in the proof of the uniqueness theorem similar to ?aginjan’s one. The constructed examples of functions show that the results cannot be improved.