Normal Families Of Meromorphic Functions Research Articles

Radically distributed value and normal families of meromorphic functions

We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.

Normal families of meromorphic functions sharing a holomorphic function

Let k be a positive integer, and m be an even number. Suppose that a(z)(≠0) is a holomorphic function with zeros of multiplicity m in a domain D. Let F be a family of meromorphic functions in a domain D such that each ƒ є F have zeros of multiplicity at least k + 1 + m and poles of multiplicity at least m + 1. It is mainly proved that for each pair (ƒ, g) є F, if ff(k) and ff(k) share a(z) IM, then F is normal in D. This result improves Hu and Meng's results published in Journal of Mathematical Analysis and Applications (2009, 2011), and also Jiang and Gao's result in Acta Matematica Scientia (2012).

Normal Families concerning Polynomials and Shared Values

We study the problem of normal families of meromorphic functions concerning polynomials and shared values. We prove that a family ℱ of meromorphic functions in a domain D is normal if, for each function f∈ℱ, Pfzfkz=a⇔fkz=b, where P is a polynomial with the origin as zero, k is a positive integer, and a ≠0, b are two finite constants.

Normal families of meromorphic functions sharing one function

Suppose p(z) is a holomorphic function, the multiplicity of its zeros is at most d, P(z) is a nonconstant polynomial. Let ℱ be a family of meromorphic functions in a domain D, all of whose zeros and poles have multiplicity at least max{ k 2 +d+1,k+d}. If for each pair of functions f and g in ℱ, P(f) f ( k ) and P(g) g ( k ) share a holomorphic function p(z), then ℱ is normal in D. It generalizes and extends the results of Jiang, Gao and Wu, Xu.MSC:30D35, 30D45.

Bicomplex Analogue of Zalcman Lemma

In this paper a bicomplex analogue of Zalcman’s heuristic principle on normal families of meromorphic functions is obtained and as an application a bicomplex version of Lappan’s five-point theorem is proved.

Normal families of meromorphic functions concerning shared values

In this article, we prove the following normality criterion: let n ≥ 2, m, k be three positive integers, and a be a non-zero complex number. Let ℱ be a family of meromorphic functions in a domain D such that each f ∈ ℱ has only zeros of multiplicity at least max{k, 2}. If for each pair of f and g in ℱ, f m (f (k)) n and g m (g (k)) n share the value a IM, then ℱ is normal in D. This extends Hu and Meng's result.

Normal Families of Meromorphic Functions Concerning Higher Derivative and Shared Values

We study the normal families related to a Hayman conjecture of higher derivative and concerning shared values and get two normal criteria. Our results improve the related theorems which were obtained independently, respectively by Fang and Yuan (2001), Yuan et al. ((2011) and (2012)), Wang et al. (2011), and Qiu et al. (2012). Meanwhile, some examples are given to show the sharpness of our results.

Normality criteria concerning sharing holomorphic functions

In this paper, we study the problem of normal families of meromorphic functions that share holomorphic functions with their linear differential polynomials. Meanwhile, some results about normal family are derived which improve and generalize several related theorems obtained by Chang, Fang and Zalcman [2], Pang [5] and Schwick [9].

Normality criteria concerning sharing holomorphic functions

In this paper, we study the problem of normal families of meromorphic functions that share holomorphic functions with their linear differential polynomials. Meanwhile, some results about normal family are derived which improve and generalize several related theorems obtained by Chang, Fang and Zalcman [2], Pang [5] and Schwick [9].

Normal families of meromorphic functions sharing a holomorphic function and the converse of the Bloch principle

In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterexample to the converse of the Bloch principle based on the theorems.

On normality of meromorphic functions with multiple zeros and sharing values

In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain D \subseteq C and n be a positive integer. Let a, b be two finite complex constants such that a \neq 0. If n \geq 3 and f + a(f')^n and g + a(g')^n share b in D for every pair of functions f, g \in F, then F is normal in D. And some examples are provided to show the result is sharp.

Normal families of meromorphic functions sharing values or functions

In this paper, we investigate the normal families of meromorphic functions concerning shared values and shared analytic functions and prove some normal criteria that generalize or extend some results obtained by Q. C. Zhang, Y. T. Li and Y. X. Gu, J. M. Chang. Mathematics Subject Classification (2000): 30D45; 30D35.

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 of meromorphic functions and shared values

Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let \({\mathcal{F}}\) be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in \({\mathcal{F}, P(f)f^{(k)}}\) and P(g)g(k) share b in D, then \({\mathcal{F}}\) is normal in D.

Shared sets and normal families of meromorphic functions

Shared sets and normal families of meromorphic functions

On normal families of meromorphic functions

In this paper, we consider some normality criterion concerning shared values. Let F be a family of meromorphic functions defined in a domain D . Let k , n ⩾ k + 2 be positive integers and a ≠ 0 be a finite complex number. If ( f n ) ( k ) and ( g n ) ( k ) share a in D for every pair of functions f , g ∈ F , then F is normal in D .

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.

Normal families of meromorphic mappings of several complex variables into CPnfor moving hypersurfaces

Normal families of meromorphic mappings of several complex variables into CPⁿfor moving hypersurfaces

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.

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 ′ − a f n and g ′ − a g n 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 .