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 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.     

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.     

Weighted sharing of meromorphic functions relative to small functions

Using the idea of weighted sharing, we prove some results on uniqueness of meromorphic functions with three weighted sharing values. The results in this paper improve those given by H.X. Yi, T.C. Alzahary and H.X. Yi, T.C. Alzahary, I. Lahiri and N. Mandal, and other authors.     

Normal criteria for families of meromorphic functions

By using the Nevanlinna theory, we prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family.     

The uniqueness theorems of meromorphic functions sharing three values and one pair of values

In this paper, we deal with a uniqueness theorem of two meromorphic functions that have three weighted sharing values and one pair of values. The results in this paper improve those given by G.G. Gundersen, G. Brosch, T.C. Alzahary, T.C. Alzahary and H.X. Yi, I. Lahiri and P. Sahoo, and other authors.     

Normal families of meromorphic functions with multiple zeros

Let F be a family of meromorphic functions defined in a domain D such that for each f∈F, all zeros of f(z) are of multiplicity at least 3, and all zeros of f^′(z) are of multiplicity at least 2 in D. If for each f∈F, f^′(z)−1 has at most 1 zero in D, ignoring multiplicity, then F is normal in D.     

Normal families and shared values of meromorphic functions

Let be a positive integer, let F be a family of meromorphic functions in a domain D, all of whose zeros have multiplicity at least k+1, and let , be two holomorphic functions on D. If, for each f∈F, f=a(z)⇔f(k)=h(z), then F is normal in D.     

The uniqueness theorems of meromorphic functions sharing three values and a set with two elements

In this paper, we deal with a uniqueness theorem of two meromorphic functions that have three weighted sharing values and a sharing set with two elements. The results in this paper improve those given by G. Brosch, K. Tohge, T.C. Alzahary and H.X. Yi and other authors.     

Normal families of meromorphic functions with multiple values

Let F be a family of meromorphic functions defined in a domain D, let ψ(?0) be a holomorphic function in D, and k be a positive integer. Suppose that, for every function f∈F, f≠0, f(k)≠0, and all zeros of f(k)−ψ(z) have multiplicities at least (k+2)/k. If, for k=1, ψ has only zeros with multiplicities at most 2, and for k?2, ψ has only simple zeros, then F is normal in D. This improves and generalizes the related results of Gu, Fang and Chang, Yang, Schwick, et al.     

Two meromorphic functions on annuli sharing some pairs of values

In this article, we prove that two admissible meromorphic functions $f$ and $g$ on an annulus must be linked by a Möbius transformation if they share a pair of values ignoring multiplicities and share other four pairs of values with multiplicities truncated by 2. We also show that two admissible meromorphic functions which share $q(q\ge 6)$ pairs of values ignoring multiplicities are linked by a Möbius transformation. Moreover, in our results, the zeros with multiplicities more than a certain number are not needed to be counted in the sharing pairs of values condition of meromorphic functions.     

On the characteristics of meromorphic functions with three weighted sharing values

In this paper, we deal with the relation between the characteristic function of two nonconstant meromorphic functions with three weighted sharing values, which improves a result given by H.X. Yi and Y.H. Li. From this we establish a theorem which improves a result given by P. Li and C.C. Yang.     

On the uniqueness theorems of meromorphic functions with weighted sharing of three values

In this paper, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions with their first derivatives having the same fixed points with the same multiplicities. The results in this paper improve those given by K. Tohge, Xiao-Min Li and Hong-Xun Yi.     

Exceptional functions and normal families of meromorphic functions with multiple zeros

Let k be a positive integer with k?2; let h(?0) be a holomorphic function which has no simple zeros in D; and let F be a family of meromorphic functions defined in D, all of whose poles are multiple, and all of whose zeros have multiplicity at least k+1. If, for each function f∈F, f(k)(z)≠h(z), then F is normal in D.     

On normal families of meromorphic functions

In this paper, we study the normality of a family of meromorphic functions and obtain some normality results for meromorphic functions, which improve and generalize the related results of Gu, Bergweiler and Lin.     

Uniqueness of meromorphic functions concerning sharing two small functions with their derivatives

In this paper, we study the uniqueness of meromorphic functions that share two small functions with their derivatives. We prove the following result: Let $f$ be a nonconstant meromorphic function such that $\mathop {\overline{\lim}}\limits_{r\to\infty} \frac{\bar{N}(r,f)}{T(r,f)}<\frac{3}{128}$, and let $a$, $b$ be two distinct small functions of $f$ with $a\not\equiv\infty$ and $b\not\equiv\infty$. If $f$ and $f"$ share $a$ and $b$ IM, then $f\equiv f"$.     

A normal criterion about two families of meromorphic functions concerning shared values

In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D■C,a1, a2, a3, a4 be four distinct finite complex numbers. If G is normal, and for every f ∈ F , there exists g ∈ G such that f(z) and g(z) share the values a1, a2, a3, a4, then F is normal on D.     

Weighted sharing three values and uniqueness of meromorphic functions

Using the idea of weighted sharing, we prove some results on uniqueness of meromorphic functions sharing three values which improve some results given by H.-X. Yi, I. Lahiri, X. Hua and other authors.     

Normal families of meromorphic functions concerning shared values

In this paper we study the problem of normal families of meromorphic functions concerning shared values and prove that a family F of meromorphic functions in a domain D is normal if for each pair of functions f and g in F, f^′−afn and g^′−agn share a value b in D where n is a positive integer and a,b are two finite constants such that n?4 and a≠0. This result is not true when n?3.