Some further results on meromorphic functions that share two sets

Qi Han; Hong-Xun Yi

Displaying similar documents to “Some further results on meromorphic functions that share two sets”

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).

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

Xiao-Tian Bai, Qi Han (2007)

Archivum Mathematicum

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This paper studies the unicity of meromorphic(resp. entire) functions of the form $f^{n} f^{'}$ and obtains the following main result: Let $f$ and $g$ be two non-constant meromorphic (resp. entire) functions, and let $a \in ℂ ∖ {0}$ be a non-zero finite value. Then, the condition that $E_{3)} (a, f^{n} f^{'}) = E_{3)} (a, g^{n} g^{'})$ implies that either $f = d g$ for some $(n + 1)$ -th root of unity $d$ , or $f = c_{1} e^{c z}$ and $g = c_{2} e^{- c z}$ for three non-zero constants $c$ , $c_{1}$ and $c_{2}$ with ${(c_{1} c_{2})}^{n + 1} c^{2} = - a^{2}$ provided that $n \geq 11$ (resp. $n \geq 6$ ). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed. ...

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.

On unique range sets of meromorphic functions in $ℂ^{m}$

Xiao-Tian Bai, Qi Han (2007)

Archivum Mathematicum

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By considering a question proposed by F. Gross concerning unique range sets of entire functions in $ℂ$ , we study the unicity of meromorphic functions in $ℂ^{m}$ that share three distinct finite sets CM and obtain some results which reduce $5 \leq c_{3} (ℳ (ℂ^{m})) \leq 9$ to $5 \leq c_{3} (ℳ (ℂ^{m})) \leq 6$ .

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.

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.

Uniqueness of meromorphic functions sharing a meromorphic function of a smaller order with their derivatives

Xiao-Min Li, Hong-Xun Yi (2010)

Annales Polonici Mathematici

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We prove some uniqueness theorems for meromorphic functions and their derivatives that share a meromorphic function whose order is less than those of the above meromorphic functions. The results in this paper improve those given by G. G. Gundersen & L. Z. Yang, J. P. Wang, J. M. Chang & Y. Z. Zhu, and others. Some examples are provided to show that our results are the best possible.

Unicity theorems for meromorphic functions that share three values

Wei-Ran Lü, Hong-Xun Yi (2003)

Annales Polonici Mathematici

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We deal with the problem of uniqueness of meromorphic functions sharing three values, and obtain several results which improve and extend some theorems of M. Ozawa, H. Ueda, H. X. Yi and other authors. We provide examples to show that results are sharp.

On meromorphic functions for sharing two sets and three sets inm-punctured complex plane

Hong-Yan Xu, Xiu-Min Zheng, Hua Wang (2016)

Open Mathematics

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In this article, we study the uniqueness problem of meromorphic functions in m-punctured complex plane Ω and obtain that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 9, such that any two admissible meromorphic functions f and g in Ω must be identical if f, g share S1, S2 I M in Ω.

Uniqueness of meromorphic functions sharing two finite sets

Jun-Fan Chen (2017)

Open Mathematics

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We prove uniqueness theorems of meromorphic functions, which show how two meromorphic functions are uniquely determined by their two finite shared sets. This answers a question posed by Gross. Moreover, some examples are provided to demonstrate that all the conditions are necessary.

Uniqueness of meromorphic functions sharing three values

Indrajit Lahiri, Arindam Sarkar (2005)

Annales Polonici Mathematici

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We prove a result on the uniqueness of meromorphic functions sharing three values with weights and as a consequence of this result we improve a recent result of W. R. Lü and H. X. Yi.

On uniqueness of meromorphic functions sharing three values and a set consisting of two small meromorphic functions

Xiao-Min Li, Hong-Xun Yi (2009)

Annales Polonici Mathematici

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We deal with a uniqueness theorem of two meromorphic functions that share three values with weights and also share a set consisting of two small meromorphic functions. Our results improve those by G. Brosch, I. Lahiri & P. Sahoo, T. C. Alzahary & H. X. Yi, P. Li & C. C. Yang, and others.

The uniqueness of meromorphic functions ink-punctured complex plane

Hong Yan Xu, San Yang Liu (2017)

Open Mathematics

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The main purpose of this paper is to investigate the uniqueness of meromorphic functions that share two finite sets in the k-punctured complex plane. It is proved that there exist two sets S1, S2 with ♯S1 = 2 and ♯S2 = 5, such that any two admissible meromorphic functions f and g in Ω must be identical if EΩ(Sj, f) = EΩ(Sj, g)(j = 1,2).