On unicity of meromorphic functions due to a result of Yang - Hua

Xiao-Tian Bai; Qi Han

Displaying similar documents to “On unicity of meromorphic functions due to a result of Yang - Hua”

Normality criteria and multiple values II

Yan Xu, Jianming Chang (2011)

Annales Polonici Mathematici

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Let ℱ be a family of meromorphic functions defined in a domain D, let ψ (≢ 0, ∞) be a meromorphic function in D, and k be a positive integer. If, for every f ∈ ℱ and z ∈ D, (1) f≠ 0, $f^{(k)} \neq 0$ ; (2) all zeros of $f^{(k)} - ψ$ have multiplicities at least (k+2)/k; (3) all poles of ψ have multiplicities at most k, then ℱ is normal in D.

Some further results on meromorphic functions that share two sets

Qi Han, Hong-Xun Yi (2008)

Annales Polonici Mathematici

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This paper concerns the uniqueness of meromorphic functions and shows that there exists a set S ⊂ ℂ of eight elements such that any two nonconstant meromorphic functions f and g in the open complex plane ℂ satisfying $E_{3)} (S, f) = E_{3)} (S, g)$ and Ē(∞,f) = Ē(∞,g) are identical, which improves a result of H. X. Yi. Also, some other related results are obtained, which generalize the results of G. Frank, E. Mues, M. Reinders, C. C. Yang, H. X. Yi, P. Li, M. L. Fang and H. Guo, and others.

Uniqueness of entire functions concerning difference polynomials

Chao Meng (2014)

Mathematica Bohemica

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In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f (z)$ and $g (z)$ be two transcendental entire functions of finite order, and $α (z)$ a small function with respect to both $f (z)$ and $g (z)$ . Suppose that $c$ is a non-zero complex constant and $n \geq 7$ (or $n \geq 10$ ) is an integer. If $f^{n} (z) (f (z) - 1) f (z + c)$ and $g^{n} (z) (g (z) - 1) g (z + c)$ share “ $(α (z), 2)$ ” (or ${(α (z), 2)}^{*}$ ), then $f (z) \equiv g (z)$ . Our results extend and generalize some well known previous...

Fixed points of meromorphic functions and of their differences and shifts

Zong-Xuan Chen (2013)

Annales Polonici Mathematici

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Let f(z) be a finite order transcendental meromorphic function such that λ(1/f(z)) < σ(f(z)), and let c ∈ ℂ∖0 be a constant such that f(z+c) ≢ f(z) + c. We mainly prove that $m a x τ (f (z)), τ (Δ_{c} f (z)) = m a x τ (f (z)), τ (f (z + c)) = m a x τ (Δ_{c} f (z)), τ (f (z + c)) = σ (f (z))$ , where τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).

Distribution of zeros and shared values of difference operators

Jilong Zhang, Zongsheng Gao, Sheng Li (2011)

Annales Polonici Mathematici

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We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that $Δ_{c}^{k} f \neq 0$ for n ≥ 2, and a is a small function with respect to f, then $f ⁿ Δ_{c}^{k} f$ equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

On the uniqueness problem for meromorphic mappings with truncated multiplicities

Feng Lü (2014)

Annales Polonici Mathematici

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The purpose of this paper is twofold. The first is to weaken or omit the condition $d i m f^{- 1} (H_{i} \cap H_{j}) \leq m - 2$ for i ≠ j in some previous uniqueness theorems for meromorphic mappings. The second is to decrease the number q of hyperplanes $H_{j}$ such that f(z) = g(z) on $⋃_{j = 1}^{q} f^{- 1} (H_{j})$ , where f,g are meromorphic mappings.

Normal families and shared values of meromorphic functions

Mingliang Fang, Lawrence Zalcman (2003)

Annales Polonici Mathematici

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Let ℱ be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d ≠ b,0 and c ≠ a. If, for each f ∈ ℱ, $f (z) = a \Leftrightarrow f^{(k)} (z) = b$ , and $f^{(k)} (z) = d \Rightarrow f (z) = c$ , then ℱ is a normal family on D. The same result holds for k=1 so long as b≠(m+1)d, m=1,2,....

On the value distribution of differential polynomials of meromorphic functions

Yan Xu, Huiling Qiu (2010)

Annales Polonici Mathematici

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Let f be a transcendental meromorphic function of infinite order on ℂ, let k ∈ ℕ and $φ = R e^{P}$ , where R ≢ 0 is a rational function and P is a polynomial, and let $a ₀, a ₁, . . ., a_{k - 1}$ be holomorphic functions on ℂ. If all zeros of f have multiplicity at least k except possibly finitely many, and $f = 0 \Leftrightarrow f^{(k)} + a_{k - 1} f^{(k - 1)} + \dots + a ₀ f = 0$ , then $f^{(k)} + a_{k - 1} f^{(k - 1)} + \dots + a ₀ f - φ$ has infinitely many zeros.

Normality criteria for families of zero-free meromorphic functions

Jun-Fan Chen (2015)

Annales Polonici Mathematici

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Let ℱ be a family of zero-free meromorphic functions in a domain D, let n, k and m be positive integers with n ≥ m+1, and let a ≠ 0 and b be finite complex numbers. If for each f ∈ ℱ, $f^{m} + a (f^{(k)}) ⁿ - b$ has at most nk zeros in D, ignoring multiplicities, then ℱ is normal in D. The examples show that the result is sharp.

Finiteness problem for meromorphic mappings sharing n+3 hyperplanes of ℙⁿ(ℂ)

Si Duc Quang (2014)

Annales Polonici Mathematici

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We prove some finiteness theorems for differential nondegenerate meromorphic mappings of $ℂ^{m}$ into ℙⁿ(ℂ) which share n+3 hyperplanes.

Multiple values and uniqueness problem for meromorphic mappings sharing hyperplanes

Ting-Bin Cao, Kai Liu, Hong-Zhe Cao (2013)

Annales Polonici Mathematici

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The purpose of this article is to deal with multiple values and the uniqueness problem for meromorphic mappings from $ℂ^{m}$ into the complex projective space ℙⁿ(ℂ) sharing hyperplanes. We obtain two uniqueness theorems which improve and extend some known results.

On certain subclasses of multivalently meromorphic close-to-convex maps

K. S. Padmanabhan (1998)

Annales Polonici Mathematici

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Let Mₚ denote the class of functions f of the form $f (z) = 1 / z^{p} + \sum_{k = 0}^{\infty} a ₖ z^{k}$ , p a positive integer, in the unit disk E = |z| < 1, f being regular in 0 < |z| < 1. Let $L_{n, p} (α) = f : f \in M ₚ, R e - (z^{p + 1} / p) {(D ⁿ f)}^{'} > α$ , α < 1, where $D ⁿ f = {(z^{n + p} f (z))}^{(n)} / (z^{p} n!)$ . Results on $L_{n, p} (α)$ are derived by proving more general results on differential subordination. These results reduce, by putting p =1, to the recent results of Al-Amiri and Mocanu.

Complex Oscillation Theory of Differential Polynomials

Abdallah El Farissi, Benharrat Belaïdi (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper, we investigate the relationship between small functions and differential polynomials $g_{f} (z) = d_{2} f^{''} + d_{1} f^{'} + d_{0} f$ , where $d_{0} (z)$ , $d_{1} (z)$ , $d_{2} (z)$ are entire functions that are not all equal to zero with $ρ (d_{j}) < 1$ $(j = 0, 1, 2)$ generated by solutions of the differential equation $f^{''} + A_{1} (z) e^{a z} f^{'} + A_{0} (z) e^{b z} f = F$ , where $a, b$ are complex numbers that satisfy $a b (a - b) \neq 0$ and $A_{j} (z) \neg \equiv 0$ ( $j = 0, 1$ ), $F (z) \neg \equiv 0$ are entire functions such that $max \{ρ (A_{j}), j = 0, 1, ρ (F)\} < 1 .$

Entire function sharing two polynomials with its $k$ th derivative

Sujoy Majumder, Nabadwip Sarkar (2024)

Mathematica Bohemica

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We investigate the uniqueness problem of entire functions that share two polynomials with their $k$ th derivatives and obtain some results which improve and generalize the recent result due to Lü and Yi (2011). Also, we exhibit some examples to show that the conditions of our results are the best possible.

Nonlinear differential monomials sharing two values

Sujoy Majumder (2016)

Mathematica Bohemica

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Using the notion of weighted sharing of values which was introduced by Lahiri (2001), we deal with the uniqueness problem for meromorphic functions when two certain types of nonlinear differential monomials namely $h^{n} h^{(k)}$ $(h = f, g)$ sharing a nonzero polynomial of degree less than or equal to $3$ with finite weight have common poles and obtain two results. The results in this paper significantly rectify, improve and generalize the results due to Cao and Zhang (2012).