scholarly journals Normality Criteria of Meromorphic Functions That Share a Holomorphic Function

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jie Ding ◽  
Jianming Qi ◽  
Taiying Zhu
Keyword(s):  
Positive Integer ◽  
Holomorphic Function ◽  
Meromorphic Functions ◽  
Holomorphic Functions

LetFbe a family of meromorphic functions defined inD, letψ(≢0),a0,a1,...,ak-1be holomorphic functions inD, and letkbe a positive integer. Suppose that, for every functionf∈F,f≠0,P(f)=f(k)+ak-1f(k-1)+⋯+a1f'+a0f≠0and, for every pair functions(f,g)∈F,P(f),P(g)shareψ, thenFis normal inD.

scholarly journals A normal criterion concerning zero numbers

2021 ◽  
Author(s):  
Chengxiong Sun
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Normal Criterion

AbstractLet $$n \ge 4$$ n ≥ 4 be a positive integer, $$\mathcal {F}$$ F be a family of meromorphic functions in D and let $$a(z)(\not \equiv 0), b(z)$$ a ( z ) ( ≢ 0 ) , b ( z ) be two holomorphic functions in D. If, for any function $$f \in \mathcal { F}$$ f ∈ F , (1)$$f(z) \ne \infty $$ f ( z ) ≠ ∞ when $$a(z)=0$$ a ( z ) = 0 , (2) $$f'(z)-a(z)f^{n}(z)-b(z)$$ f ′ ( z ) - a ( z ) f n ( z ) - b ( z ) has at most one zero in D, then $$\mathcal {F}$$ F is normal in D.

Exceptional functions and normal families of holomorphic functions with multiple zeros

2011 ◽  
Vol 18 (1) ◽  
pp. 31-38
Author(s):  
Jun-Fan Chen
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Normal Families ◽  
Multiple Zeros ◽  
Zeros Of Functions

Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

1970 ◽  
Vol 22 (4) ◽  
pp. 803-814 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Value Distribution ◽  
Minimum Modulus ◽  
Image Position

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

scholarly journals NORMAL FAMILIES OF ZERO-FREE MEROMORPHIC FUNCTIONS

2011 ◽  
Vol 91 (3) ◽  
pp. 313-322 ◽  
Author(s):  
BINGMAO DENG ◽  
MINGLIANG FANG ◽  
DAN LIU
Keyword(s):  
Positive Integer ◽  
Holomorphic Function ◽  
Meromorphic Functions ◽  
Normal Families

AbstractLet ℱ be a family of zero-free meromorphic functions in a domain D, let h be a holomorphic function in D, and let k be a positive integer. If the function f(k)−h has at most k distinct zeros (ignoring multiplicity) in D for each f∈ℱ, then ℱ is normal in D.

scholarly journals A Criterion for Normalcy

1968 ◽  
Vol 32 ◽  
pp. 277-282 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Meromorphic Function ◽  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions

Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals NORMALITY AND SHARED SETS

2009 ◽  
Vol 86 (3) ◽  
pp. 339-354 ◽  
Author(s):  
MINGLIANG FANG ◽  
LAWRENCE ZALCMAN
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Complex Numbers ◽  
Finite Complex ◽  
Shared Sets ◽  
Zeros Of Functions

AbstractLet ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

scholarly journals Projective varieties and rings of Thetanullwerte

1990 ◽  
Vol 118 ◽  
pp. 165-176
Author(s):  
Riccardo Salvati Manni
Keyword(s):  
Positive Integer ◽  
Holomorphic Function ◽  
Half Space ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Integral Closure ◽  
Graded Ring ◽  
Projective Varieties ◽  
Entry Vector

Let r denote an even positive integer, m an element of Q2g such that r·m ≡ 0 mod 1 and ϑm the holomorphic function on the Siegel upper-half space Hg defined by(1) ,in which e(t) stands for exp and m′ and m″ are the first and the second entry vector of m. Let Θg(r) denote the graded ring generated over C by such Thetanullwerte; then it is a well known fact that the integral closure of Θg(r) is the ring of all modular forms relative to Igusa’s congruence subgroup Γg(r2, 2r2) cf. [6]. We shall denote this ring by A(Γg(r2, 2r2)).

scholarly journals Boundary behaviour of holomorphic functions on the cardioid domain with some applications

2019 ◽  
Vol 38 (7) ◽  
pp. 203-218
Author(s):  
Shatha Sami Alhily ◽  
_ Deepmala
Keyword(s):  
General Solution ◽  
Holomorphic Function ◽  
Unit Disk ◽  
Laplace Equation ◽  
Holomorphic Functions ◽  
Research Paper ◽  
Boundary Behaviour ◽  
Polar Function

The objective of this research paper is to show how the Bennan'sconjecture  become a useful tool  to construct a holomorphic function on the cardioid domain, and on the boundary of unit disk. Moreover , we have addressed some applications on the existence of cusp on the boundary of arising from integrability of conformalmaps through one of the polar function in the general solution of Laplace equation.

scholarly journals ON THE ANNIHILATOR OF A DOLBEAULT GROUP

2010 ◽  
Vol 21 (03) ◽  
pp. 317-331
Author(s):  
IMRE PATYI
Keyword(s):  
Holomorphic Function ◽  
Open Subset ◽  
Cohomology Group ◽  
Holomorphic Functions ◽  
Dolbeault Cohomology ◽  
Infinite Dimensional ◽  
Continuous Character

We show that any Dolbeault cohomology group Hp,q(D), p ≥ 0, q ≥ 1, of an open subset D of a closed finite codimensional complex Hilbert submanifold of ℓ2 is either zero or infinite dimensional. We also show that any continuous character of the algebra of holomorphic functions of a closed complex Hilbert submanifold M of ℓ2 is induced by its evaluation at a point of M. Lastly, we prove that any closed split infinite dimensional complex Banach submanifold of ℓ2 admits a nowhere critical holomorphic function.

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