Displaying similar documents to “Fixed points of meromorphic functions and of their differences and shifts”

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.

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 ∈ 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 ≥ 11 (resp. n ≥ 6 ). It improves a result of C. C. Yang and X. H. Hua. Also, some other related problems are discussed. ...

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 ) ≠ 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.

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 ≤ c 3 ( ℳ ( ℂ m ) ) ≤ 9 to 5 ≤ c 3 ( ℳ ( ℂ m ) ) ≤ 6 .

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 ≠ 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.

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.

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 ∩ H j ) ≤ 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.

Results on the deficiencies of some differential-difference polynomials of meromorphic functions

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

Open Mathematics

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In this paper, we study the relation between the deficiencies concerning a meromorphic function f(z), its derivative f′(z) and differential-difference monomials f(z)mf(z+c)f′(z), f(z+c)nf′(z), f(z)mf(z+c). The main results of this paper are listed as follows: Let f(z) be a meromorphic function of finite order satisfying lim sup r→+∞ T(r, f) T(r,  f ′ ) <+∞, lim sup r → + ∞ T ( r , f ) T ( r , f ' ) < + ∞ , and c be a non-zero complex constant, then δ(∞, f(z)m f(z+c)f′(z))≥δ(∞, f′) and δ(∞,f(z+c)nf′(z))≥ δ(∞, f′). We also investigate...