Distribution of zeros and shared values of difference operators

Jilong Zhang; Zongsheng Gao; Sheng Li

Displaying similar documents to “Distribution of zeros and shared values of difference operators”

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.

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.

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 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.

On zeros of differences of meromorphic functions

Yong Liu, HongXun Yi (2011)

Annales Polonici Mathematici

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Let f be a transcendental meromorphic function and $g (z) = f (z + c ₁) + \dots + f (z + c_{k}) - k f (z)$ and $g_{k} (z) = f (z + c ₁) \dots f (z + c_{k}) - f^{k} (z)$ . A number of results are obtained concerning the exponents of convergence of the zeros of g(z), $g_{k} (z)$ , g(z)/f(z), and $g_{k} (z) / f^{k} (z)$ .

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.

Some Results on the Properties of Differential Polynomials Generated by Solutionsof Complex Differential Equations

Zinelâabidine LATREUCH, Benharrat BELAÏDI (2015)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $f^{(k)} + A_{k - 1} (z) f^{(k - 1)} + \dots + A_{1} (z) f^{'} + A_{0} (z) f = 0,$ where $A_{i} (z)$ $(i = 0, 1, \dots, k - 1)$ are meromorphic functions of finite order in the complex plane.

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. ...

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).

Universal sequences for Zalcman’s Lemma and $Q_{m}$ -normality

Shahar Nevo (2005)

Annales Polonici Mathematici

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We prove the existence of sequences ${ϱ ₙ}_{n = 1}^{\infty}$ , ϱₙ → 0⁺, and ${z ₙ}_{n = 1}^{\infty}$ , |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F (z) = F_{G, α} (z)$ on ℂ such that $ϱ ₙ^{α} F (n z ₙ + n ϱ ₙ ζ)$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_{m}$ -normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

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.

Pull-back of currents by meromorphic maps

Tuyen Trung Truong (2013)

Bulletin de la Société Mathématique de France

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Let $X$ and $Y$ be compact Kähler manifolds, and let $f : X \to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{♯}$ for currents of bidegrees $(p, p)$ of finite order on $Y$ (and thus forcurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony,...

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...

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.

Some subclasses of meromorphic and multivalent functions

Ding-Gong Yang, Jin-Lin Liu (2014)

Annales Polonici Mathematici

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The authors introduce two new subclasses $F_{p, k} (λ, A, B)$ and $G_{p, k} (λ, A, B)$ of meromorphically multivalent functions. Distortion bounds and convolution properties for $F_{p, k} (λ, A, B)$ , $G_{p, k} (λ, A, B)$ and their subclasses with positive coefficients are obtained. Some inclusion relations for these function classes are also given.

Nonlinear differential polynomials sharing a non-zero polynomial with finite weight

Abhijit Banerjee, Molla Basir AHAMED (2016)

Mathematica Bohemica

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In the paper, dealing with a question of Lahiri (1999), we study the uniqueness of meromorphic functions in the case when two certain types of nonlinear differential polynomials, which are the derivatives of some typical linear expression, namely $h^{n} {(h - 1)}^{m}$ ( $h = f, g$ ), share a non-zero polynomial with finite weight. The results obtained in the paper improve, extend, supplement and generalize some recent results due to Sahoo (2013), Li and Gao (2010). In particular, we have shown that under a suitable...