Growth of solutions of a class of complex differential equations

Ting-Bin Cao

Displaying similar documents to “Growth of solutions of a class of complex differential equations”

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.

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 $max \{ρ (A_{j}), j = 0, 1, ρ (F)\} < 1 .$

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 $r : \sum_{1 \leq ν \leq k} l o g ⁺ m a x_{| | z | | = r} {| f (z) - q_{ν} (z) |}^{- 1} < B (γ) T (r, f)$ .

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

David Adam (2004)

Acta Arithmetica

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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.

Value distribution and uniqueness of difference polynomials and entire solutions of difference equations

Xiaoguang Qi (2011)

Annales Polonici Mathematici

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This paper is devoted to value distribution and uniqueness problems for difference polynomials of entire functions such as fⁿ(f-1)f(z+c). We also consider sharing value problems for f(z) and its shifts f(z+c), and improve some recent results of Heittokangas et al. [J. Math. Anal. Appl. 355 (2009), 352-363]. Finally, we obtain some results on the existence of entire solutions of a difference equation of the form $f ⁿ + P (z) {(Δ_{c} f)}^{m} = Q (z) .$

Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

Antonio Garijo, Xavier Jarque, Mónica Moreno Rocha (2011)

Fundamenta Mathematicae

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We consider the family of transcendental entire maps given by $f_{a} (z) = a (z - (1 - a)) e x p (z + a)$ where a is a complex parameter. Every map has a superattracting fixed point at z = -a and an asymptotic value at z = 0. For a > 1 the Julia set of $f_{a}$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of...

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.

Continuation of holomorphic functions with growth conditions and some of its applications

Alexander V. Abanin, Pham Trong Tien (2010)

Studia Mathematica

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We prove a generalization of the well-known Hörmander theorem on continuation of holomorphic functions with growth conditions from complex planes in $ℂ^{p}$ into the whole $ℂ^{p}$ . We apply this result to construct special families of entire functions playing an important role in convolution equations, interpolation and extension of infinitely differentiable functions from closed sets. These families, in their turn, are used to study optimal or canonical, in a certain sense, weight sequences defining...

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.

On the approximation of entire functions over Carathéodory domains

Devendra Kumar, Harvir S. Kasana (1994)

Commentationes Mathematicae Universitatis Carolinae

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Let $D$ be a Carathéodory domain. For $1 \leq p \leq \infty$ , let $L^{p} (D)$ be the class of all functions $f$ holomorphic in $D$ such that ${∥ f ∥}_{D, p} = [\frac{1}{A} \int \int_{D}^{} {| f (z) |}^{p} {d x d y]}^{1 / p} < \infty$ , where $A$ is the area of $D$ . For $f \in L^{p} (D)$ , set $E_{n}^{p} (f) = inf_{t \in π_{n}} {∥ f - t ∥}_{D, p};$ $π_{n}$ consists of all polynomials of degree at most $n$ . In this paper we study the growth of an entire function in terms of approximation error in $L^{p}$ -norm on $D$ .

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.

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

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

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In this Note II we continue the analysis of the phenomenon of mechanical twinning that we began in a preceding Note I $(^{1})$ . Furthermore, we point out some fundamental properties useful in the study of growth twins, for which a fully comprehensive thermoelastic theory is not yet available.

Uniqueness of entire functions concerning difference polynomials

Chao Meng (2014)

Mathematica Bohemica

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In this paper, we investigate the uniqueness problem of difference polynomials sharing a small function. With the notions of weakly weighted sharing and relaxed weighted sharing we prove the following: Let $f (z)$ and $g (z)$ be two transcendental entire functions of finite order, and $α (z)$ a small function with respect to both $f (z)$ and $g (z)$ . Suppose that $c$ is a non-zero complex constant and $n \geq 7$ (or $n \geq 10$ ) is an integer. If $f^{n} (z) (f (z) - 1) f (z + c)$ and $g^{n} (z) (g (z) - 1) g (z + c)$ share “ $(α (z), 2)$ ” (or ${(α (z), 2)}^{*}$ ), then $f (z) \equiv g (z)$ . Our results extend and generalize some well known previous...

On the growth of entire functions of n complex variables

Małgorzata Downarowicz, Adam Janik (1985)

Annales Polonici Mathematici

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Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

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

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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”.