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

Zinelâabidine LATREUCH; Benharrat BELAÏDI

Displaying similar documents to “Some Results on the Properties of Differential Polynomials Generated by Solutionsof Complex Differential Equations”

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 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,....

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 the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

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Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

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.

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

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.

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 $Δ_{c}^{k} f \neq 0$ 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.

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${P_{i}}_{i = 1}^{\infty}$ generated by a three-term recurrence relation $P_{i} (x) + Q_{1} (x) P_{i - 1} (x) + Q_{2} (x) P_{i - 2} (x) = 0$ with the standard initial conditions $P_{0} (x) = 1, P_{- 1} (x) = 0,$ where $Q_{1} (x)$ and $Q_{2} (x)$ are arbitrary real polynomials.

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...

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 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).

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...

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...

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.

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...

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.

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.