Generalizations on the results of Cao and Zhang

Sujoy Majumder; Rajib Mandal

Displaying similar documents to “Generalizations on the results of Cao and Zhang”

Nonlinear differential monomials sharing two values

Sujoy Majumder (2016)

Mathematica Bohemica

Similarity:

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).

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

Abhijit Banerjee, Molla Basir AHAMED (2016)

Mathematica Bohemica

Similarity:

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...

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

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

Mathematica Bohemica

Similarity:

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.

Uniqueness of meromorphic functions concerning value sharing of nonlinear differential monomials

Sujoy Majumder, Rajib Mandal (2020)

Mathematica Bohemica

Similarity:

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 the value distribution of differential polynomials of meromorphic functions

Yan Xu, Huiling Qiu (2010)

Annales Polonici Mathematici

Similarity:

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.

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

Duc Quang Si, An Hai Tran (2020)

Mathematica Bohemica

Similarity:

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...

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

Bappaditya Bhowmik, Sambhunath Sen (2024)

Czechoslovak Mathematical Journal

Similarity:

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.

Normal families and shared values of meromorphic functions

Mingliang Fang, Lawrence Zalcman (2003)

Annales Polonici Mathematici

Similarity:

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 certain subclasses of multivalently meromorphic close-to-convex maps

K. S. Padmanabhan (1998)

Annales Polonici Mathematici

Similarity:

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.

Normality criteria and multiple values II

Yan Xu, Jianming Chang (2011)

Annales Polonici Mathematici

Similarity:

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 uniqueness problem for meromorphic mappings with truncated multiplicities

Feng Lü (2014)

Annales Polonici Mathematici

Similarity:

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.

Meromorphic function sharing a small function with a linear differential polynomial

Indrajit Lahiri, Amit Sarkar (2016)

Mathematica Bohemica

Similarity:

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...

On the generalization of two results of Cao and Zhang

Sujoy Majumder (2017)

Mathematica Bohemica

Similarity:

This paper studies the uniqueness of meromorphic functions $f^{n} \prod_{i = 1}^{k} {(f^{(i)})}^{n_{i}} and g^{n} \prod_{i = 1}^{k} {(g^{(i)})}^{n_{i}}$ that share two values, where $n, n_{k}, k \in ℕ$ , $n_{i} \in ℕ \cup {0}$ , $i = 1, 2, ..., k - 1$ . The results 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

Similarity:

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.

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

Sujoy Majumder, Lata Mahato (2023)

Mathematica Bohemica

Similarity:

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

Similarity:

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,...

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

Abdelkader Dahmani, Benharrat Belaidi (2025)

Mathematica Bohemica

Similarity:

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...

Uniqueness results for differential polynomials sharing a set

Soniya Sultana, Pulak Sahoo (2025)

Mathematica Bohemica

Similarity:

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).

On zeros of differences of meromorphic functions

Yong Liu, HongXun Yi (2011)

Annales Polonici Mathematici

Similarity:

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