Finite Complex Numbers Research Articles

Normal Criteria of Function Families Concerning Shared Values

We study the normality of families of meromorphic functions concerning shared values. We consider whether a family of meromorphic functions ℱ is normal in D, if, for every pair of functions f and g in ℱ, f′ − af−n and g′ − ag−n share the value b, where a and b are two finite complex numbers such that a ≠ 0, n is a positive integer. Some examples show that the conditions in our results are best possible.

A Note on Normal Families of Meromorphic Functions Concerning Shared Values

We study the normality of families of meromorphic functions related to a Hayman conjecture. We consider whether a family of meromorphic functions ℱ is normal in D if, for every pair of functions f and g in ℱ, f′ − afn and g′ − agn share the value b for n = 1, 2, and 3, where a and b ≠ 0 are two finite complex numbers. Some examples show that the conditions in our results are the best possible.

Normality and shared values concerning differential polynomials

Let \( \mathcal{F} \) be a family of functions meromorphic in a domain D, let P be a polynomial with either deg P ⩾ 3 or deg P = 2 and P having only one distinct zero, and let b be a finite nonzero complex number. If, each pair of functions f and g in \( \mathcal{F} \), P(f)f′ and P(g)g′ share b in D, then \( \mathcal{F} \) is normal in D.

Some Normal Criteria about Shared Values with Their Multiplicity Zeros

Let ℱ be a family of meromorphic functions in the domain D, all of whose zeros are multiple. Let n (n ≥ 2) be an integer and let a, b be two nonzero finite complex numbers. If f + a(f′) n and g + a(g′) n share b in D for every pair of functions f, g ∈ ℱ, then ℱ is normal in D.

Picard values and normal families of meromorphic functions

Let f be a transcendental meromorphic function on the complex plane ℂ, let a be a non-zero finite complex number and let n and k be two positive integers. In this paper, we prove that if n≥k+1, then $\smash{f+a(f^{(k)})^n}$ assumes each value b∈ℂ infinitely often. Also, the related normal criterion for families of meromorphic functions is given. Our results generalize the related results of Fang and Zalcman.

NORMALITY AND SHARED SETS

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.

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 .

Some results on the regular solutions of a linear differential equation

In this paper, we prove a theorem on the regular growth of the solutions of a linear differential equation, as an application of which we obtain some uniqueness theorems of an entire function sharing a finite nonzero complex number with its n th derivative and a linear differential polynomial of its n th derivative under the restriction of finite lower order or of finite lower hyper-order that is not a positive integer. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.

On the value distribution of ƒ + a(ƒ′) n

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) n assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given.

Entire functions sharing one polynomial with their derivatives

In this paper, we study the growth of solutions of ak-th order linear differential equation and that of a k + 1-th order linear differential equation. From this we affirmatively answer a uniqueness question concerning a conjecture given by Bruck in 1996 under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM. The results in this paper also improve some known results. Some examples are provided to show that the results in this paper are best possible.

Normal families and shared values of holomorphic functions

Let {ie335-1} be a family of holomorphic functions in a domain D, k be a positive integer, a, b(≠0), c(≠0) and d be finite complex numbers. If, for each {ie335-2}, all zeros of f−d have multiplicity at least k, f( k )=a whenever f=0, and f=c whenever f( k )=b, then {ie335-3} is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.

Meromorphic functions that weighted sharing three values and one pair

G. Brosch improved the theorem of Nevanlinna for four values theorem and proved that let f and g be two nonconstant meromorphic functions sharing 0, 1, ∞ CM, and let a and b be two finite complex numbers such that $a,b\not\in \{0,1\}$. If $f=a\Leftrightarrow g=b$, then f is a fractional linear transformation of g. In this paper we extend this theorem by using the idea of weighted sharing.

Normal families of holomorphic functions and shared values

LetF be a family of holomorphic functions in a domainD, and leta, b be two distinct finite complex numbers. If, for anyf ∈F,f andf′ sharea IM, andf′(z)=b wheneverf(z)=b, thenF is normal inD. This improves results due to Pang, Pang and Zalcman, Xu, etc.

On a class of starlike functions defined in a halfplane

Let D = {z: Re z < 0} and let S*(D) be the class of univalent functions normalized by the conditions $lim_{D ∋ z → ∞}(f(z) - z) = a$, a a finite complex number, 0 ∉ f(D), and mapping D onto a domain f(D) starlike with respect to the exterior point w = 0.

On permutability of periodic entire functions

On donne les fonctions permutables avec une fonction entiere periodique de type exponentiel

Hayman directions of meromorphic functions

In this paper, we shall prove the existence of the singular directions related to Hayman's problems[1]. The results are as follows. Theorem I. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≠0, −1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$ Theorem II. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$ Theorem III. Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$

PROOF OF HAYMAN'S CONJECTURE ON NORMAL FAMILIES

In 1964, Hayman posed the following conjecture. Let a(≠0) and b be two finite complex numbers and suppose n(≥5) be a positive integer. If is a family of meromorphic functions in a domain D and for each f∈ and z∈D, there exists f'(z)—af(z) n ≠b, then is normal in D. This paper aims at giving a proof of the conjecture.

Deficient values of entire functions and their derivatives

Let f ( z ) f(z) be entire and of finite order, f ( n ) {f^{(n)}} be the n n th derivative, and Δ n ( f ) = Σ δ ( a , f ( n ) ) {\Delta _n}(f) = \Sigma \delta (a,{f^{(n)}}) , the sum of all deficient values of f ( n ) {f^{(n)}} . The authors show that Δ n ( f ) {\Delta _n}(f) can be strictly increasing.