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

Sujoy Majumder; Nabadwip Sarkar

Displaying similar documents to “Entire function sharing two polynomials with its $k$ th derivative”

Uniqueness of entire functions and fixed points

Xiao-Guang Qi, Lian-Zhong Yang (2010)

Annales Polonici Mathematici

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Let f and g be entire functions, n, k and m be positive integers, and λ, μ be complex numbers with |λ| + |μ| ≠ 0. We prove that ${(f ⁿ (z) (λ f^{m} (z) + μ))}^{(} k)$ must have infinitely many fixed points if n ≥ k + 2; furthermore, if ${(f ⁿ (z) (λ f^{m} (z) + μ))}^{(} k)$ and ${(g ⁿ (z) (λ g^{m} (z) + μ))}^{(} k)$ have the same fixed points with the same multiplicities, then either f ≡ cg for a constant c, or f and g assume certain forms provided that n > 2k + m* + 4, where m* is an integer that depends only on λ.

Uniqueness results for differential polynomials sharing a set

Soniya Sultana, Pulak Sahoo (2025)

Mathematica Bohemica

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We investigate the uniqueness results of meromorphic functions if differential polynomials of the form ${(Q (f))}^{(k)}$ and ${(Q (g))}^{(k)}$ share a set counting multiplicities or ignoring multiplicities, where $Q$ is a polynomial of one variable. We give suitable conditions on the degree of $Q$ and on the number of zeros and the multiplicities of the zeros of $Q^{'}$ . The results of the paper generalize some results due to T. T. H. An and N. V. Phuong (2017) and that of N. V. Phuong (2021).

A note on some results of Li and Li

Sujoy Majumder, Somnath Saha (2018)

Mathematica Bohemica

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The purpose of the paper is to study the uniqueness problems of linear differential polynomials of entire functions sharing a small function and obtain some results which improve and generalize the related results due to J. T. Li and P. Li (2015). Basically we pay our attention to the condition $λ (f) \neq 1$ in Theorems 1.3, 1.4 from J. T. Li and P. Li (2015). Some examples have been exhibited to show that conditions used in the paper are sharp.

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.

Uniqueness of meromorphic functions concerning value sharing of nonlinear differential monomials

Sujoy Majumder, Rajib Mandal (2020)

Mathematica Bohemica

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With the idea of normal family we study the uniqueness of meromorphic functions $f$ and $g$ when $f^{n} {(f^{(k)})}^{m} - p$ and $g^{n} {(g^{(k)})}^{m} - p$ share two values, where $p$ is any nonzero polynomial. The result of this paper significantly improves and generalizes the result due to A. Banerjee and S. Majumder (2018).

Car-Pólya and Gel’fond’s theorems for $_{q} [T]$

David Adam (2004)

Acta Arithmetica

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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.

Bounds for the derivative of certain meromorphic functions and on meromorphic Bloch-type functions

Bappaditya Bhowmik, Sambhunath Sen (2024)

Czechoslovak Mathematical Journal

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It is known that if $f$ is holomorphic in the open unit disc $𝔻$ of the complex plane and if, for some $c > 0$ , $| f (z) | \leq {1 / (1 - | z |}^{2})^{c}$ , $z \in 𝔻$ , then $| f^{'} {(z) | \leq 2 (c + 1) / (1 - | z |}^{2})^{c + 1}$ . We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in $𝔻$ . In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.

Meromorphic function sharing a small function with a linear differential polynomial

Indrajit Lahiri, Amit Sarkar (2016)

Mathematica Bohemica

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The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and...

Finiteness of meromorphic functions on an annulus sharing four values regardless of multiplicity

Duc Quang Si, An Hai Tran (2020)

Mathematica Bohemica

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This paper deals with the finiteness problem of meromorphic funtions on an annulus sharing four values regardless of multiplicity. We prove that if three admissible meromorphic functions $f_{1}$ , $f_{2}$ , $f_{3}$ on an annulus $𝔸 (R_{0})$ share four distinct values regardless of multiplicity and have the of positive counting function, then $f_{1} = f_{2}$ or $f_{2} = f_{3}$ or $f_{3} = f_{1}$ . This result deduces that there are at most two admissible meromorphic functions on an annulus sharing a value with multiplicity truncated to level $2$ and sharing...

Finite logarithmic order meromorphic solutions of linear difference/differential-difference equations

Abdelkader Dahmani, Benharrat Belaidi (2025)

Mathematica Bohemica

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Firstly we study the growth of meromorphic solutions of linear difference equation of the form $A_{k} (z) f (z + c_{k}) + \dots + A_{1} (z) f (z + c_{1}) + A_{0} (z) f (z) = F (z),$ where $A_{k} (z), ..., A_{0} (z)$ and $F (z)$ are meromorphic functions of finite logarithmic order, $c_{i}$ $(i = 1, ..., k, k \in ℕ)$ are distinct nonzero complex constants. Secondly, we deal with the growth of solutions of differential-difference equation of the form $\sum_{i = 0}^{n} \sum_{j = 0}^{m} A_{i j} (z) f^{(j)} (z + c_{i}) = F (z),$ where $A_{i j} (z)$ $(i = 0, 1, ..., n, j = 0, 1, ..., m, n, m \in ℕ)$ and $F (z)$ are meromorphic functions of finite logarithmic order, $c_{i}$ $(i = 0, ..., n)$ are distinct complex constants. We extend some previous results...

Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

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We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{(k)} + a_{k - 1} (z) f^{(k - 1)} + \dots + a ₁ (z) f^{'} + D (z) f = 0$ , (1) where $D (z) = Q ₁ (z) e^{P ₁ (z)} + Q ₂ (z) e^{P ₂ (z)} + Q ₃ (z) e^{P ₃ (z)}$ , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), $a_{j} (z)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

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 the meromorphic solutions of a certain type of nonlinear difference-differential equation

Sujoy Majumder, Lata Mahato (2023)

Mathematica Bohemica

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The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $f^{n} (z) + P_{d} (z, f) = p_{1} (z) e^{α_{1} (z)} + p_{2} (z) e^{α_{2} (z)},$ where $P_{d} (z, f)$ is a difference-differential polynomial in $f (z)$ of degree $d \leq n - 1$ with small functions of $f (z)$ as its coefficients, $p_{1}$ , $p_{2}$ are nonzero rational functions and $α_{1}$ , $α_{2}$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

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,...

Generalizations on the results of Cao and Zhang

Sujoy Majumder, Rajib Mandal (2022)

Mathematica Bohemica

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We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P (f) \prod_{i = 1}^{k} {(f^{(i)})}^{n_{i}}$ and $P (g) \prod_{i = 1}^{k} {(g^{(i)})}^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.

Uniqueness of $q$ -shift difference polynomials of meromorphic functions sharing a small function

Renukadevi S. Dyavanal, Rajalaxmi V. Desai (2020)

Mathematica Bohemica

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We investigate the uniqueness of a $q$ -shift difference polynomial of meromorphic functions sharing a small function which extend the results of N. V. Thin (2017) to $q$ -difference operators.

Displaying similar documents to “Entire function sharing two polynomials with its k th derivative”

Displaying similar documents to “Entire function sharing two polynomials with its $k$ th derivative”