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

Abdelkader Dahmani; Benharrat Belaidi

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Displaying similar documents to “Finite logarithmic order meromorphic solutions of linear difference/differential-difference equations”

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.

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.

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...

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,...

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.

Uniqueness and differential polynomials of meromorphic functions sharing a nonzero polynomial

Pulak Sahoo (2016)

Mathematica Bohemica

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Let $k$ be a nonnegative integer or infinity. For $a \in ℂ \cup {\infty}$ we denote by $E_{k} (a; f)$ the set of all $a$ -points of $f$ where an $a$ -point of multiplicity $m$ is counted $m$ times if $m \leq k$ and $k + 1$ times if $m > k$ . If $E_{k} (a; f) = E_{k} (a; g)$ then we say that $f$ and $g$ share the value $a$ with weight $k$ . Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to...

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).

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 $g_{k} (z) / f^{k} (z)$ .

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...

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.

Twists and resonance of $L$ -functions, I

Jerzy Kaczorowski, Alberto Perelli (2016)

Journal of the European Mathematical Society

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We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1 / d$ of the $L$ -functions of any degree $d \geq 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

Maximally convergent rational approximants of meromorphic functions

Hans-Peter Blatt (2015)

Banach Center Publications

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Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_{ρ (f)}$ , ρ(f) < ∞. We investigate rational approximants $r_{n, m ₙ}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $ρ {(f)}^{- n}$ on E implies uniform maximal convergence in m₁-measure inside $E_{ρ (f)}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_{ρ (f)}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue...

A remark on complex powers of analytic functions

Giuseppe Zampieri (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Sia $K\subset\subset\mathbb{R}^{n}$ un compatto, $f\geq 0$ una funzione analitica all'intorno di $K$ , ed $m$ la massima molteplicità in $K$ degli zeri di $f$ ; si prova che la potenza $f^{\lambda}$ ( $\lambda\in\mathbb{C}$ , $Re\lambda>—\frac{1}{m}$ ) è integrabile in $K$ . L'estensione meromorfa dell'applicazione $\lambda\rightarrow f^{\lambda}$ da $Re\lambda>0$ a tutto $\mathbb{C}$ (con valori in $\mathcal{D}^{\prime}(K)$ anziché in $L^{1}(K)$ ) era già stata provata in [1] e [2].

Global behavior of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}$

Raafat Abo-Zeid (2015)

Archivum Mathematicum

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In this paper, we introduce an explicit formula and discuss the global behavior of solutions of the difference equation $x_{n + 1} = \frac{a x_{n - 3}}{b + c x_{n - 1} x_{n - 3}}, n = 0, 1, \dots$ where $a, b, c$ are positive real numbers and the initial conditions $x_{- 3}$ , $x_{- 2}$ , $x_{- 1}$ , $x_{0}$ are real numbers.