Showing papers on "Entire function published in 2023"
28 Feb 2023
TL;DR: In this paper , the authors examined complex functions as transformations and showed that complex functions can be visualized as points in the complex plane, so that a complex function f becomes a transformation of the plane.
Abstract: Abstract This chapter examines complex functions as transformations. A complex function f is a rule that assigns to a complex number z an image complex number w = f(z). In order to investigate such functions, it is essential that one is able to visualize them. Several methods exist for doing this, but the chapter focuses almost exclusively on the method introduced in the previous chapter. This means viewing z and its image w as points in the complex plane, so that f becomes a transformation of the plane. The chapter begins by looking at polynomials, power series, the exponential function, and cosine and sine. It then considers multifunctions, the logarithm function, and the process of averaging over circles.
31 Jan 2023
TL;DR: In this article , the authors describe a general method to generate superoscillatory functions of several variables starting from a super-oscillating sequence of one variable, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire.
Abstract: In this paper we describe a general method to generate superoscillatory functions of several variables starting from a superoscillating sequence of one variable. Our results are based on the study of suitable infinite order differential operators on holomorphic functions with growth conditions of exponential type, where additional constraints are required when dealing with infinite order differential operators whose symbol is a function that is holomorphic in some open set, but not necessarily entire. The results proved for the superoscillating sequence in several variables are extended to sequences of supershifts in several variables.
TL;DR: In this paper , the uniqueness of entire functions concerning their difference operator and derivatives was studied and a non-trivial proof of this result using Nevanlinna's value distribution theory was given.
Abstract: In this paper we study the uniqueness of entire functions concerning their difference operator and derivatives. The idea of entire and meromorphic functions relies heavily on this direction. Rubel and Yang considered the uniqueness of entire function and its derivative and proved that if $f(z)$ and $f'(z)$ share two values $a,b$ counting multilicities then $f(z)\equiv f'(z)$. Later, Li Ping and Yang improved the result given by Rubel and Yang and proved that if $f(z)$ is a non-constant entire function and $a,b$ are two finite distinct complex values and if $f(z)$ and $f^{(k)}(z)$ share $a$ counting multiplicities and $b$ ignoring multiplicities then $f(z)\equiv f^{(k)}(z)$. In recent years, the value distribution of meromorphic functions of finite order with respect to difference analogue has become a subject of interest. By replacing finite distinct complex values by polynomials, we prove the following result: Let $\Delta f(z)$ be trancendental entire functions of finite order, $ k \geq 0$ be integer and $P_{1}$ and $P_{2}$ be two polynomials. If $\Delta f(z)$ and $f^{(k)}$ share $P_{1}$ CM and share $P_{2}$ IM, then $\Delta f \equiv f^{(k)}$. A non-trivial proof of this result uses Nevanlinna's value distribution theory.
TL;DR: In this article , the authors develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree.
Abstract: We develop techniques that lay out a basis for generalizations of the famous Thurston's Topological Characterization of Rational Functions for an infinite set of marked points and branched coverings of infinite degree. Analogously to the classical theorem we consider the Thurston's $\sigma$-map acting on a Teichm\"uller space which is this time infinite-dimensional -- and this leads to a completely different theory comparing to the classical setting. We demonstrate our techniques by giving an alternative proof of the result by Markus F\"orster about the classification of exponential functions with the escaping singular value.
14 Mar 2023
TL;DR: In this article , it was shown that if k (≥ 3) is an integer, for the given non-negative real number λ (may be infinity), then there exists an entire function ƒ(z) which with its linear differential polynomial share 1 CM, such that the k -th order of ǫ(ǫ) is λ .
Abstract: Abstract In this paper, we discuss the values of k -th order of the entire functions which with its linear differential polynomial share 1 CM. We prove that if k (≥ 3) is an integer, for the given non-negative real number λ (may be infinity), then there exists an entire function ƒ(z) which with its linear differential polynomial share 1 CM, such that the k -th order of ƒ(z) is λ . MSC Classification: 30D35 , 30D20
05 Jun 2023
TL;DR: In this paper , it was shown that any subset of the Euclidean space with rational coefficients is the exceptional set of uncountable many transcendental entire functions over the space.
Abstract: In this paper, among other things, we prove that any subset of $\overline{\mathbb{Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountable many transcendental entire functions over $\mathbb{C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions.
TL;DR: In this paper , the eigenvalues of a loaded differential operator of the first order with a periodic boundary condition on the interval [1; 1], that is, equation contains a load at the point (1) and the function of bounded variation (t), with the condition Φ(−1) = ǫ( 1) = 1 .
Abstract: In the paper, we consider the problem on eigenvalues of a loaded differential operator of the first order with a periodic boundary condition on the interval [–1; 1], that is, equation contains a load at the point (–1) and the function of bounded variation (t), with the condition Φ(−1) = Φ(1) = 1 . A characteristic determinant of spectral problem is constructed for the considered loaded differentiation operator, which is an entire analytical function on the spectral parameter. On the basis of the characteristic determinant formula, conclusions are proved about the asymptotic behavior of the spectrum and eigenfunctions of the loaded spectral problem for the differentiation operator, the characteristic determinant of which is an entire analytic function of the spectral parameter l. A theorem on the location of eigenvalues on the complex plane l is formulated, where the regular growth of an entire analytic function is indicated. A theorem is proved on the asymptotics of the zeros of an entire function, that is, the eigenvalues of the original considered spectral problem for a loaded differential operator of differentiation, and the asymptotic properties of an entire function with distribution of roots are studied.
TL;DR: In this article , the Borel lemma and Clunie lemma were used to deduce the relationship between an entire function f of hyper-order less than 1 and its n-th difference operator if they share a finite set and f has a Borel exceptional value 0.
Abstract: In this paper, due to the Borel lemma and Clunie lemma, we will deduce the
relationship between an entire function f of hyper-order less than 1 and its
n-th difference operator ?nc f (z) if they share a finite set and f has a
Borel exceptional value 0, where the set consists of two entire functions of
smaller orders. Moreover, the exact form of f is given and an example is
provided to show the sharpness of the condition.
13 Feb 2023
TL;DR: In this paper , the authors considered vector valued reproducing kernel Hilbert spaces (RKHS) of entire functions associated with operator valued kernel functions and derived the complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable.
Abstract: In this paper, we have considered vector valued reproducing kernel Hilbert spaces (RKHS) $\mathcal{H}$ of entire functions associated with operator valued kernel functions. de Branges operators $\mathfrak{E}=(E_- , E_+)$ analogous to de Branges matrices have been constructed with the help of pairs of Fredholm operator valued entire functions on $\mathfrak{X}$, where $\mathfrak{X}$ is a complex seperable Hilbert space. A few explicit examples of these de Branges operators are also discussed. The newly defined RKHS $\mathcal{B}(\mathfrak{E})$ based on the de Branges operator $\mathfrak{E}=(E_-,E_+)$ has been characterized under some special restrictions. The complete parametrizations and canonical descriptions of all selfadjoint extensions of the closed, symmetric multiplication operator by the independent variable have been given in terms of unitary operators between ranges of reproducing kernels. A sampling formula for the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been discussed. A particular class of entire operators with infinite deficiency indices has been dealt with and shown that they can be considered as the multiplication operator for a specific class of these de Branges spaces. Finally, a brief discussion on the connection between the characteristic function of a completely nonunitary contraction operator and the de Branges spaces $\mathcal{B}(\mathfrak{E})$ has been given.
16 May 2023
TL;DR: In this article , the existence of transcendental entire functions having a property studied by Mahler, namely that a function can be dense in the set of all entire transcendental functions, is proved.
Abstract: We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.
10 Apr 2023
TL;DR: In this paper , the authors construct universal entire curves in projective spaces whose Nevanlinna characteristic functions grow slower than any preassigned transcendental growth rate, and they can make such curves to be hypercyclic for translation operations along any given countable directions.
Abstract: We construct explicit universal entire curves in projective spaces whose Nevanlinna characteristic functions grow slower than any preassigned transcendental growth rate. Moreover, we can make such curves to be hypercyclic for translation operations along any given countable directions.
28 Feb 2023
TL;DR: In this paper , the authors provide a relatively short and condensed account of the topological dynamics of all meromorphic functions with an emphasis on Fatou domains, including Baker domains that are exclusive for transcendental functions and do not occur for rational functions.
Abstract: In this chapter, we provide a relatively short and condensed account of the topological dynamics of all meromorphic functions with an emphasis on Fatou domains, including Baker domains that are exclusive for transcendental functions and do not occur for rational functions. We also carry out a thorough analysis of the singular set of the inverse of a meromorphic function and all its iterates. In particular, we study at length asymptotic values and their relations to transcendental tracts. The results of this analysis will be used very frequently to study the topological structure of connected components (and their boundaries) of Fatou sets in this part of the book and a countless number of times when we move on to dealing with elliptic functions.
09 May 2023
TL;DR: In this article , it was shown that the Taylor partial sums tend to infinity in norm for lower-triangular summability methods such as the Cesaro means, and it can even be shown that it is possible to show that the sequence of partial sums is unbounded in norm.
Abstract: It is known that there exist functions in certain de Branges--Rovnyak spaces whose Taylor series diverge in norm, even though polynomials are dense in the space. This is often proved by showing that the sequence of Taylor partial sums is unbounded in norm. In this note we show that it can even happen that the Taylor partial sums tend to infinity in norm. We also establish similar results for lower-triangular summability methods such as the Ces\`aro means.
TL;DR: In this paper , it was shown that an entire function f of order ρ ∈ (0;+∞) with zeros on a finite system of rays Γm is a function of improved regular growth if and only if for some ρ 2 ∈ ρ and every q ∈ [1; +∞, one has
Abstract: Let f be an entire function, f(0) = 1, F(z) = zf′ (z)/f(z), and Γm = $$\bigcup_{j=1}^{m}\left\{z:\mathrm{arg}z={\psi }_{j}\right\},$$ 0 ≤ ψ1 < ψ2 < ...< ψm < 2π. An entire function f is called a function of improved regular growth if for some ρ ∈ (0;+∞) and ρ2 ∈ (0; ρ), and a 2π-periodic ρ-trigonometrically convex function h(φ) ≢ −∞, there exists a set U ⊂ $${\mathbb{C}}$$ contained in the union of disks with finite sum of radii such that $$\begin{array}{cc}\mathrm{log}\left|f\left(z\right)\right|={\left|z\right|}^{\rho }h\left(\varphi \right)+o\left({\left|z\right|}^{\rho 2}\right),& U
ot
i z={re}^{i\varphi }\to \infty .\end{array}$$ In this paper, we prove that an entire function f of order ρ ∈ (0;+∞) with zeros on a finite system of rays Γm is a function of improved regular growth if and only if for some ρ2 ∈ (0; ρ) and every q ∈ [1;+∞), one has $$\begin{array}{cc}{\left\{\frac{1}{2\pi }{\int }_{0}^{2\pi }{\left|\frac{F\left({re}^{i\varphi }\right)}{{r}^{\rho }}|-\rho \widetilde{h}\left(\varphi \right)\right|}^{q}d\varphi \right\}}^{1/q}=o\left({r}^{\rho 2-\rho }\right),& r\to +\infty ,\end{array}$$ where $$\widetilde{h}$$ (φ) = h(φ) − ih′ (φ)/ρ and h(φ) is the indicator of the function f.
04 Jul 2023
TL;DR: In this paper , it was shown that the exponent of convergence of non-trivial solutions of higher-order linear differential equations is infinite for any non trivial solution of $y(k)}+A_1(z)y = 0.
Abstract: We study higher order linear differential equation $y^{(k)}+A_1(z)y=0$ with $k\geq2$, where $A_1=A+h$, $A$ is a transcendental entire function of finite order with $\frac{1}{2}\leq \mu(A)<1$ and $h
eq0$ is an entire function with $\rho(h)<\mu(A)$. Then it is shown that, if $f^{(k)}+A(z)f=0$ has a solution $f$ with $\lambda(f)<\mu(A)$ then exponent of convergence of zeros of any non trivial solutions of $y^{(k)}+A_1(z)y=0$ is infinite.
TL;DR: In this paper , the authors studied the higher order delay differential equations and obtained necessary conditions on the degree of R(z,w) for them to admit a subnormal transcendental meromorphic solution.
Abstract: We study the higher order delay differential equationsw(z+1)−w(z−1)+a(z)w(k)(z)w(z)=R(z,w(z)), andw(z+1)+a(z)w(k)(z)w(z)=R(z,w(z)), where k is a positive integer, a(z) is a rational function and R(z,w) is rational in w with rational coefficients. We obtain necessary conditions on the degree of R(z,w) for these delay differential equations to admit a subnormal transcendental meromorphic solution. These results generalize some of the previous results by Halburd and Korhonen (2017) [7] to higher order case. Some examples are given to support our conclusions.
TL;DR: In this article , the class XE of all complex Banach spaces X such that X ⊇ ε was considered and the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.
Abstract: Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.
11 Jan 2023
TL;DR: Omega functions as discussed by the authors generalize Euler Gamma functions and study the functional difference equation they satisfy under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation is finite dimensional.
Abstract: We introduce Omega functions that generalize Euler Gamma functions and study the functional difference equation they satisfy. Under a natural exponential growth condition, the vector space of meromorphic solutions of the functional equation is finite dimensional. We construct a basis of the space of solutions composed by Omega functions. Omega functions are defined as exponential periods. They have a meromorphic extension to the complex plane of order $1$ with simple poles at negative integers. They are characterized by their growth property on vertical strips and their functional equation. This generalizes Wielandt's characterization of Euler Gamma function. We also introduce Incomplete Omega functions that play an important role in the proofs.
11 Apr 2023
TL;DR: In this paper , two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carath-eodory--Fej\'er--Tur\'an problem, were studied.
Abstract: We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carath\'eodory--Fej\'er--Tur\'an problem. The first variation imposes the additional requirement that the function is radially decreasing while the second one is a generalization which involves derivatives of the entire function. Various interesting inequalities, inspired by results due to Duffin and Schaeffer, Landau, and Hardy and Littlewood, are also established.