In this paper we shall consider the assymptotic growth of $|P_n(z)|^{1/k_n}$ where $P_n(z)$ is a sequence of entire functions of genus zero. Our results extend a result of J. Muller and A. Yavrian. We shall prove that if the sequence of entire functions has a geometric growth at each point in a set $E$ being non-thin at $\infty$ then it has a geometric growth in $\CC$ also. Moreover, if $E$ has some more properties, a similar result also holds for a more general kind of growth. Even in the case where $P_n$ are polynomials, our results are new in the sense that it does not require $k_n\succeq deg(P_n)$ as usually required.
Abstract. In this work we use an elementary method to derive an upper bound on the right half-plane for genus 0 entire functions if it has only negative zeros. The bound only uses information of the function on the positive real axis. Applications to the modified Bessel functions and the Airy function are also provided.
We expand the classical balayage of measures and subharmonic functions on a system of rays $S$ with a common origin on the complex plane $\mathbb C$. This allows for an arbitrary subharmonic function $v$ of finite order on $\mathbb C$ build $δ$-subharmonic function on $\mathbb C$, harmonic outside $S$ coinciding with $v$ on $S$ outside the polar set, of the same order of growth as $v$. We give applications to the relationship of the growth of an entire function in $S$ with the distribution of its zeros.
A Gauss-Lucas theorem is proved for multivariate entire functions, using a natural notion of separate convexity to obtain sharp results. Previous work in this area is mostly restricted to univariate entire functions (of genus no greater than one unless "realness" assumptions are made.) The present work applies to multivariate entire functions whose one dimensional sections can be written as a monomial times a canonical product of arbitrary genus. Essential use is made of the Levy-Steinitz theorem for conditionally convergent real number series.
We deduce some growth properties of composite entire functions in the light of their relative $(p,q)$\ th order by extending some results of J. Tu, Z. X. Chen and X. M. Zheng.
In this paper we wish to prove some results related to the growth rates of entire and meromorphic functions on the basis of relative (p,q) th order and relative (p,q) th type of a meromorphic function with respect to an entire function for any two positive integers p and q.
We investigate the conjugate indicator diagram or, equivalently, the indicator function of (frequently) hypercyclic functions of exponential type for differential operators. This gives insights into growth conditions of these functions on particular rays or sectors. Our research extends known results in several respects.
We derive in the closed and unimprovable form the bilateral non-asymptotic relations between growth of entire functions and decay rate at infinity of its Taylor coefficients. We investigate the functions of one as well as of several complex variables. We will apply the convex analysis: Young-Fenchel (Legendre) transform, Young inequality, saddle-point method etc.
We consider transcendental entire solutions of linear $q$-difference equations with polynomial coefficients and determine the asymptotic behavior of their Taylor coefficients. We use this to show that under a suitable hypothesis on the associated Newton-Puiseux diagram their zeros are asymptotic to finitely many geometric progressions. We also sharpen previous results on the growth rate of entire solutions.
Benjamini and Schramm introduced the notion of distributional limit of a sequence of graphs with uniformly bounded valence and studied such limits in the case that the involved graphs are planar. We investigate distributional limits of sequences of Riemannian manifolds with bounded curvature which satisfy certain condition of quasi-conformal nature. We then apply our results to somewhat improve Benjamini's and Schramm's original result on the recurrence of the simple random walk on limits of planar graphs. For instance, as an application give a proof of the fact that for graphs in an expander …
In this paper we wish to study some growth properties of entire functions represented by a vector valued Dirichlet series on the basis of (p,q)th relative Ritt order, (p,q)th relative Ritt type and (p,q)th relative Ritt weak type.
In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of entire functions of bounded orders $P_n(z)$, we found a nearly optimal condition, given in terms of zeros of $P_n$, for which $(k_n)$ that we have \begin{eqnarray*} \limsup_{n\to\infty}|P_n(z)|^{1/k_n}\leq 1 \end{eqnarray*} for all $z\in \mathbb C$ (see Theorem \ref{theo5}). Exploring the growth of a sequence of entire functions of bounded orders lead natura…
We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given $p \in [1,\infty ]$ and $b \in (0,a)$, where $a = \frac{1}{2 max\{2,p\}}$, we prove that there exists a distributionally irregular entire function $f$ for the operator D such that its p-integral mean function $M_p(f,r)$ grows not more rapidly than $e^r r^{-b}$. This completes related known results about the possible rates of growth of such means for D-hypercyclic entire functions. It is also obtained the existence of dense linear submanifolds of H(C) …
Let $A_p(\C)$ be the space of entire functions such that $| f(z)|\le Ae^{Bp(z)}$ for some $A,B>0$ and let $V$ be a discrete sequence of complex numbers which is not a uniqueness set for $A_p(\C)$. We use $L^2$ estimates for the $\bar\partial$ equation to charaterize the trace of $A_p(\C)$ on $V$.
In this paper we aim to establish some results depending on the comparative growth properties of composite transcendental entire or meromorphic functions and some special type of differential polynomials generated by one of the factors on the basis of (p,q)-th order and (p,q)-th lower order where p,q are positive integers with p greater than or equal to q.
In this study, we show that all non-trivial solutions of $f"+A(z)f'+B(z)f=0$ have infinite order, provided that the entire coefficient $A(z)$ has certain restrictions and $B(z)$ has multiply-connected Fatou component. We also extend these results to higher order linear differential equations.
A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume of these sets in certain polycylinders.
In this paper we define the notion of non-thin at $\infty$ as follows: Let $E$ be a subset of $\Bbb C^m$. For any $R>0$ define $E_R=E\cap \{z\in \Bbb C ^m :|z|\leq R\}$. We say that $E$ is non-thin at $\infty$ if \lim_{R\to\infty}V_{E_R}(z)=0 for all $z\in \Bbb C^m$, where $V_E$ is the pluricomplex Green function of $E$. This definition of non-thin at $\infty$ has good properties: If $E\subset \Bbb C^m$ is non-thin at $\infty$ and $A$ is pluripolar then $E\backslash A$ is non-thin at $\infty$, if $E\subset \Bbb C^m$ and $F\subset \Bbb C^n$ are closed sets non-thin at $\infty$ then $E\times F\s…
We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two.
In the second part of this work was developed a technique of balayage of finite genus $q=0,1,2,\dots$ for measures (charges) and ($δ$-) subharmonic functions of finite order to an arbitrary closed system of rays $S$ with vertex at origin on the complex plane $\mathbb C$. In this third part of our work, we use only the case $q=1$ when $S$ is a pair of oppositely directed rays, i.e., $S$ is a straight line as the point set, and balayage is made from both sides of this line. We consider measures and subharmonic functions of finite type of order $1$. This bilateral balayage of genus $1$ will be ap…
Previously, we developed the technique of balayage of measures or charges and ($δ$-)subharmonic functions of finite order onto an closed system of rays $S$ with a vertex at zero on the complex plane $\mathbb C$. In this article, we use only two kinds of balayage of measure and charge, as well as of subharmonic functions of finite type under the order $1$ and their differences. First, it is a classical balayage of the genus $q=0$ on a system of four closed rays: positive and negative, and real and imaginary semi-axis $\mathbb R^+$, $-\mathbb R^+$, $i\mathbb R^+$, $-i\mathbb R$. Second, it is tw…
In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C- action defined on standard probability space. In the same paper he asked about the minimal possible growth of measurably entire functions. In this work we show that for every arbitrary free C- action defined on a standard probability space there exists a measurably entire function whose growth does not exceed exp (exp[log^p |z|]) for any p > 3. This complements a recent result by Buhovski, Glücksam, Logunov, and Sodin (arXiv:1703.08101) who showed that such functions cann…
We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves developing a new technique to give an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, giving a sharp estimate for the growth in many cases.
We solve a problem posed by A. Bonilla and K.-G. Grosse-Erdmann by constructing an entire function $f$ that is frequently hypercyclic with respect to the differentiation operator, and satisfies $M_f(r)\leq\displaystyle ce^r r^{-1/4}$, where $c>0$ be chosen arbirarily small. The obtained growth rate is sharp. We also obtain optimal results for the growth when measured in terms of average $L^p$-norms. Among other things, the proof applies Rudin-Shapiro polynomials and heat kernel estimates.
Let f be an entire function of finite order less than 1. The maximum modulus M(r) of f and the counting function of the zeros N(r) are connected by a best possible growth inequality known as Valiron's Theorem: For functions subharmonic in d-dimensional Euclidean space, Hayman obtained a corresponding result with a best possible constant involving the dimension d. For the special case of an entire function on d-dimensional complex space, we obtain a corresponding dimension-free, best possible inequality.
Let f be a transcendental entire function in the Eremenko-Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.
We give conditions ensuring that the Julia set and the escaping set of an entire function of completely regular growth have positive Lebesgue measure. The essential hypotheses are that the indicator is positive except perhaps at isolated points and that most zeros are located in neighborhoods of finitely many rays. We apply the result to solutions of linear differential equations.
We give a complete description of zero sets for some well-known subclasses of entire functions of exponential growth (bounded on real axis, Cartwright's class)
Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.
We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.
We announce conditions under which a given sequence of points on the complex plane is a subsequence of zeros of an entire function with weight restrictions on growth.
Let $f$ be a transcendental entire function. The quite fast escaping set, $Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast escaping set, $A(f),$ under certain conditions. In this paper we generalise these sets by introducing a family of sets $Q_m(f)$, $m \in \mathbb{N}.$ We also give one regularity and one growth condition which imply that $Q_m(f)$ is equal to $A(f)$ and we show that all functions of finite order and positive lower order satisfy $Q_m(f)=A(f)$ for any $m$. Finally, we relate the new regularity condition to a sufficient condition for $Q_2(f)=A(f)$ …
If $f$ is an entire function and $a$ is a complex number, $a$ is said to be an asymptotic value of $f$ if there exists a path $γ$ from $0$ to infinity such that $f(z) - a$ tends to $0$ as $z$ tends to infinity along $γ$. The Denjoy--Carleman--Ahlfors Theorem asserts that if $f$ has $n$ distinct asymptotic values, then the rate of growth of $f$ is at least order $n/2$, mean type. A long-standing problem asks whether this conclusion holds for entire functions having $n$ distinct asymptotic (entire) functions, each of growth at most order $1/2$, minimal type. In this paper conditions on the funct…
In this paper, we will prove that all non-trivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order, where we have some restrictions on entire functions $A(z)$ and $B(z)$.
Let $g$ be a entire function of exponential type on the complex plane $\mathbb C$, $Z=\{ z_k\}_{k=1,2,\dots}$ be a sequence of points in $\mathbb C$. We give a criterion for the existence of an entire function $f\neq 0$ of exponential type that vanishes on $ Z$ and satisfies the constraint $\ln |f(iy)|\leq \ln |g(iy)|+o(|y|)$, $y\to \pm\infty$. Our results generalize and develop a joint results of P. Malliavin and L. A. Rubel. Applications to multipliers for entire functions of exponential type, to analytic functionals and their convolutions on the complex plane, as well as to the completeness…
Given a compact Riemann surface X and a point x_0 in X, we construct a holomorphic function without critical points on the punctured Riemann surface R = X - x_0 which is of finite order at the point x_0. This complements the result of Gunning and Narasimhan from 1967 who constructed a noncritical holomorphic function on every open Riemann surface, but without imposing any growth condition. On the other hand, if the genus of X is at least one, then we show that every algebraic function on R admits a critical point. Our proof also shows that every cohomology class in H^1(X;C) is represented as a…
In 1979 I. Ciorănescu and L. Zsidó have proved a minimum modulus theorem for entire functions dominated by the restriction to the positive half axis of a canonical product of genus zero, having all roots on the positive imaginary axis and satisfying a certain condition. Here we prove that the above result is optimal: if a canonical product ω of genus zero, having all roots on the positive imaginary axis, does not satisfy the condition in the 1979 paper, then always there exists an entire function dominated by the restriction to the positive half axis of ω, which does not satisfy the desired mi…
Considering differential equation f''+A(z)f'+B(z)f=0, where A(z) and B(z) are entire complex functions, our results revolve around proving all non-trivial solutions are of infinite order taking various restrictions on coefficients A(z) and B(z).
In this paper we obtain estimates for certain transcendence measures of an entire function $f$. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial $P(z,w)$ in ${\Bbb C}^2$ along the graph of $f$. These inequalities provide, in turn, estimates for the number of zeros of the function $P(z,f(z))$ in the disk of radius $r$, in terms of the degree of $P$ and of $r$. Our estimates hold for arbitrary entire functions $f$ of finite order, and for a subsequence $\{n_j\}$ of degrees of polynomials. But for special classes of functions, including the Riemann $ζ$-…
We obtain a complete description of the Riesz measures of almost periodic subharmonic functions with at most of linear growth on the complex plane; as a consequence we get a complete description of zero sets for the class of entire functions of exponential type with almost periodic modulus.
We estimate the growth of the canonical integral of Hadamard-Weierstrass of measure of finite order on the complex plane by the type of counting function or average counting function of this measure
A weighted space of entire functions rapidly decreasing on the real line is considered in the paper. A growth of these functions along the imaginary axis is controlled by some system of weight functions. The Fourier transform of functions of this space is studied. Equivalent description of the considered space in terms of estimates on derivatives of functions on real line is obtained.
Let $Z$ and $W$ be a pair of point distributions of finite upper density on the complex plane $\mathbb C$ with the real axis $\mathbb R$. We give several variants of necessary and at the same time sufficient conditions for their arrangement, under which for every entire function of exponential type $g\neq 0$ vanishing on $W$, there is, respectively, either an entire function of exponential type $f\neq 0$ vanishing on $Z$ and satisfies one of the two variants of the constraints: 1) $|f(iy)|\leq |g(iy)|$ at each $y\in \mathbb R$, i.e. everywhere on the imaginary axis $i\mathbb R$, 2) $\ln |f(iy)…
In this paper we introduce the notions of (p,q)-th relative Gol'dberg order and (p,q)-th relative Gol'dberg type of entire functions of several complex variables where p,q are any positive integers. Then we study some growth properties of entire functions of several complex variables on the basis of their (p,q)-th relative Gol'dberg order and (p,q)-th relative Gol'dberg type.
We prove a form of the $\cos πρ$ theorem which gives strong estimates for the minimum modulus of a transcendental entire function of order zero. We also prove a generalisation of a result of Hinkkanen that gives a sufficient condition for a transcendental entire function to have no unbounded Fatou components. These two results enable us to show that there is a large class of entire functions of order zero which have no unbounded Fatou components. On the other hand we give examples which show that there are in fact functions of order zero which not only fail to satisfy Hinkkanen's condition but…
In this course of lectures we give an account of the growth theory of subharmonic functions, which is directed towards its applications to entire functions of one and several complex variables.
We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions a…
We show that the fast escaping set $A(f)$ of a transcendental entire function $f$ has a structure known as a spider's web whenever the maximum modulus of $f$ grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko.
We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2.
We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show that the result holds under suitable growth conditions involving the minimum and maximum modulus.
Let $f$ be a transcendental entire function. The fast escaping set $A(f)$, various regularity conditions on the growth of the maximum modulus of $f$, and also, more recently, the quite fast escaping set $Q(f)$ have all been used to make progress on fundamental questions concerning the iteration of $f$. In this paper we establish new relationships between these three concepts. We prove that a certain weak regularity condition is necessary and sufficient for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\neq A(f)$. We also apply a result of Beurling that relates the size of the minim…
