On deviations from rational functions of entire functions of finite lower order

E. Ciechanowicz; I. I. Marchenko

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Displaying similar documents to “On deviations from rational functions of entire functions of finite lower order”

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.

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”.

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.

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 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.

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function $f$ which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number $ℓ$ of terms. Then we look at the possible decompositions $f (x) = g (h (x))$ , where $g, h$ are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of $g$ is bounded only in terms of $ℓ$ (and we provide explicit bounds). This supports and quantifies...

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}$ , where $t_{x} = x (x + 1) / 2$ is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system $t_{x} + t_{y} = t_{p}, t_{y} + t_{z} = t_{q}, t_{x} + t_{z} = t_{r}, t_{x} + t_{y} + t_{z} = t_{s}$ has infinitely many rational two-parameter solutions.

Characteristic Exponents of Rational Functions

Anna Zdunik (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider two characteristic exponents of a rational function f:ℂ̂ → ℂ̂ of degree d ≥ 2. The exponent $χ_{a} (f)$ is the average of log∥f’∥ with respect to the measure of maximal entropy. The exponent $χ_{m} (f)$ can be defined as the maximal characteristic exponent over all periodic orbits of f. We prove that $χ_{a} (f) = χ_{m} (f)$ if and only if f(z) is conformally conjugate to $z \mapsto z^{\pm d}$ .

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.

Some characterizations of the class $_{m} (Ω)$ and applications

Hai Mau Le, Hong Xuan Nguyen, Hung Viet Vu (2015)

Annales Polonici Mathematici

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We give some characterizations of the class $_{m} (Ω)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

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.

Lower semicontinuous envelopes in $W^{1, 1} \times L^{p}$

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type $\int_{Ω} f (x, u, v, \nabla u) d x$ with respect to the $(W^{1, 1} \times L^{p})$ -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1, 1} \times L^{p}$ for the lower semicontinuous envelope is also provided.

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence. ...

Entire solutions to a class of fully nonlinear elliptic equations

Ovidiu Savin (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We study nonlinear elliptic equations of the form $F (D^{2} u) = f (u)$ where the main assumption on $F$ and $f$ is that there exists a one dimensional solution which solves the equation in all the directions $ξ \in ℝ^{n}$ . We show that entire monotone solutions $u$ are one dimensional if their $0$ level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

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

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In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\mathfrak{I}(u)$ , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\mathfrak{I}(u)$ are reduced to simpler form $H(v)$ and $\mathcal{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u)\leq\mathfrak{I}(u)$ , $H(v)\leq\mathcal{H}(v)$ , we obtain the existence of the absolute minimum for the Serrin forms $\mathfrak{I}(u)$ ...

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...

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.

Extensions of the Cugiani-Mahler theorem

Yann Bugeaud (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In 1955, Roth established that if $ξ$ is an irrational number such that there are a positive real number $ε$ and infinitely many rational numbers $p / q$ with $q \geq 1$ and $| ξ - p / q | < q^{- 2 - ε}$ , then $ξ$ is transcendental. A few years later, Cugiani obtained the same conclusion with $ε$ replaced by a function $q \mapsto ε (q)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $| ξ - p / q | < q^{- 2 - ε (q)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it...

Maximal upper asymptotic density of sets of integers with missing differences from a given set

Ram Krishna Pandey (2015)

Mathematica Bohemica

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Let $M$ be a given nonempty set of positive integers and $S$ any set of nonnegative integers. Let $\bar{δ} (S)$ denote the upper asymptotic density of $S$ . We consider the problem of finding $μ (M) : = sup_{S} \bar{δ} (S),$ where the supremum is taken over all sets $S$ satisfying that for each $a, b \in S$ , $a - b \notin M .$ In this paper we discuss the values and bounds of $μ (M)$ where $M = {a, b, a + n b}$ for all even integers and for all sufficiently large odd integers $n$ with $a < b$ and $gcd (a, b) = 1 .$