A note on some results of Li and Li

Sujoy Majumder; Somnath Saha

Displaying similar documents to “A note on some results of Li and Li”

The hyper-order of solutions of certain linear complex differential equations

Guowei Zhang, Ang Chen (2010)

Annales Polonici Mathematici

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We prove some theorems on the hyper-order of solutions of the equation $f^{(k)} - e^{Q} f = a (1 - e^{Q})$ , where Q is an entire function, which is a polynomial or not, and a is an entire function whose order can be larger than 1. We improve some results by J. Wang and X. M. Li.

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.

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.

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

David Adam (2004)

Acta Arithmetica

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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 deviations from rational functions of entire functions of finite lower order

E. Ciechanowicz, I. I. Marchenko (2007)

Annales Polonici Mathematici

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Let f be a transcendental entire function of finite lower order, and let $q_{ν}$ be rational functions. For 0 < γ < ∞ let B(γ):= πγ/sinπγ if γ ≤ 0.5, B(γ):= πγ if γ > 0.5. We estimate the upper and lower logarithmic density of the set $r : \sum_{1 \leq ν \leq k} l o g ⁺ m a x_{| | z | | = r} {| f (z) - q_{ν} (z) |}^{- 1} < B (γ) T (r, f)$ .

Complex Oscillation Theory of Differential Polynomials

Abdallah El Farissi, Benharrat Belaïdi (2011)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In this paper, we investigate the relationship between small functions and differential polynomials $g_{f} (z) = d_{2} f^{''} + d_{1} f^{'} + d_{0} f$ , where $d_{0} (z)$ , $d_{1} (z)$ , $d_{2} (z)$ are entire functions that are not all equal to zero with $ρ (d_{j}) < 1$ $(j = 0, 1, 2)$ generated by solutions of the differential equation $f^{''} + A_{1} (z) e^{a z} f^{'} + A_{0} (z) e^{b z} f = F$ , where $a, b$ are complex numbers that satisfy $a b (a - b) \neq 0$ and $A_{j} (z) \neg \equiv 0$ ( $j = 0, 1$ ), $F (z) \neg \equiv 0$ are entire functions such that $max \{ρ (A_{j}), j = 0, 1, ρ (F)\} < 1 .$

Convergence of Taylor series in Fock spaces

Haiying Li (2014)

Studia Mathematica

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It is well known that the Taylor series of every function in the Fock space $F_{α}^{p}$ converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in $F_{α}^{p}$ do not necessarily converge “in norm”.

Nonexistence of entire positive solution for a conformal $k$ -Hessian inequality

Feida Jiang, Saihua Cui, Gang Li (2020)

Czechoslovak Mathematical Journal

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In this paper, we study the nonexistence of entire positive solution for a conformal $k$ -Hessian inequality in $ℝ^{n}$ via the method of proof by contradiction.

Entire functions of exponential type not vanishing in the half-plane $ℑ z > k$ , where $k > 0$

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $P (z)$ be a polynomial of degree $n$ having no zeros in $| z | < k$ , $k \leq 1$ , and let $Q (z) : = z^{n} \bar{P (1 / \bar{z})}$ . It was shown by Govil that if ${max}_{| z | = 1} | P^{'} (z) |$ and ${max}_{| z | = 1} | Q^{'} (z) |$ are attained at the same point of the unit circle $| z | = 1$ , then $max_{| z | = 1} | P^{'} (z) | \leq \frac{n}{1 + k^{n}} max_{| z | = 1} | P (z) | .$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Michal Johanis (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that a $C^{k}$ -smooth mapping on an open subset of $ℝ^{n}$ , $k \in ℕ \cup {0, \infty}$ , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.

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).

Stević-Sharma type operators on Fock spaces in several variables

Lijun Ma, Zicong Yang (2024)

Czechoslovak Mathematical Journal

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Let $ϕ$ be an entire self-map of $ℂ^{N}$ , $u_{0}$ be an entire function on $ℂ^{N}$ and $𝐮 = (u_{1}, \dots, u_{N})$ be a vector-valued entire function on $ℂ^{N}$ . We extend the Stević-Sharma type operator to the classcial Fock spaces, by defining an operator $T_{u_{0}, 𝐮, ϕ}$ as follows: $- . 4 p t T_{u_{0}, 𝐮, ϕ} f = u_{0} \cdot f \circ ϕ + \sum_{i = 1}^{N} u_{i} \cdot \frac{\partial f}{\partial z_{i}} \circ ϕ .$ We investigate the boundedness and compactness of $T_{u_{0}, 𝐮, ϕ}$ on Fock spaces. The complex symmetry and self-adjointness of $T_{u_{0}, 𝐮, ϕ}$ are also characterized.

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

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),...

A study of various results for a class of entire Dirichlet series with complex frequencies

Niraj Kumar, Garima Manocha (2018)

Mathematica Bohemica

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Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_{k} e^{〈 λ^{k}, z 〉}$ for which $(| λ^{k} {| / e)}^{| λ^{k} |} k! | a_{k} |$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

The factorization of $f (x) x^{n} + g (x)$ with $f (x)$ monic and of degree $\leq 2$ .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux

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In this paper we investigate the factorization of the polynomials $f (x) x^{n} + g (x) \in ℤ [x]$ in the special case where $f (x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f (x)$ is monic and linear.