scholarly journals On a Result of Hayman Concerning the Maximum Modulus Set

2021 ◽  
Author(s):  
Vasiliki Evdoridou ◽  
Leticia Pardo-Simón ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Taylor Series ◽  
Upper Bound ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Algebraic Condition ◽  
Exact Number ◽  
Small Set ◽  
Set Of Points ◽  
Analytic Curves

AbstractThe set of points where an entire function achieves its maximum modulus is known as the maximum modulus set. In 1951, Hayman studied the structure of this set near the origin. Following work of Blumenthal, he showed that, near zero, the maximum modulus set consists of a collection of disjoint analytic curves, and provided an upper bound for the number of these curves. In this paper, we establish the exact number of these curves for all entire functions, except for a “small” set whose Taylor series coefficients satisfy a certain simple, algebraic condition. Moreover, we give new results concerning the structure of this set near the origin, and make an interesting conjecture regarding the most general case. We prove this conjecture for polynomials of degree less than four.

On the Growth of Entire Functions Bounded on Large Sets

1977 ◽  
Vol 29 (6) ◽  
pp. 1287-1291
Author(s):  
Lowell J. Hansen
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Modulus Function ◽  
Classical Theorem ◽  
Minimum Modulus ◽  
Rate Of Growth ◽  
Large Sets ◽  
Growth Of Entire Functions ◽  
Lindelöf Theorem

There have been many indications of a relationship between the rate of growth of an entire function and the “size” of the set, E(c), where the modulus of the function is larger than the constant, c. Theorems of this type include the classical theorem of Wiman on functions of bounded minimum modulus, the Phragmén-Lindelöf Theorem, the Denjoy-Carleman-Ahlfors Theorem, and its many subsequent improvements. These theorems can all be understood as quantitative versions of the statement that if ƒ is an entire function such that, for some c > 0, the set E(c) is ‘'small”, then the maximum modulus function M(R, f) is forced to grow rapidly with R.

scholarly journals The minimal growth of entire functions with given zeros along unbounded sets

2020 ◽  
Vol 54 (2) ◽  
pp. 146-153
Author(s):  
I. V. Andrusyak ◽  
P.V. Filevych
Keyword(s):  
Entire Function ◽  
Continuous Function ◽  
Entire Functions ◽  
Counting Function ◽  
Maximum Modulus ◽  
Positive Function ◽  
Minimal Growth ◽  
Complex Sequence ◽  
Growth Of Entire Functions ◽  
Necessary And Sufficient

Let $l$ be a continuous function on $\mathbb{R}$ increasing to $+\infty$, and $\varphi$ be a positive function on $\mathbb{R}$. We proved that the condition$$\varliminf_{x\to+\infty}\frac{\varphi(\ln[x])}{\ln x}>0$$is necessary and sufficient in order that for any complex sequence $(\zeta_n)$ with $n(r)\ge l(r)$, $r\ge r_0$, and every set $E\subset\mathbb{R}$ which is unbounded from above there exists an entire function $f$ having zeros only at the points $\zeta_n$ such that$$\varliminf_{r\in E,\ r\to+\infty}\frac{\ln\ln M_f(r)}{\varphi(\ln n_\zeta(r))\ln l^{-1}(n_\zeta(r))}=0.$$Here $n(r)$ is the counting function of $(\zeta_n)$, and $M_f(r)$ is the maximum modulus of $f$.

scholarly journals Dynamics of slowly growing entire functions

2001 ◽  
Vol 63 (3) ◽  
pp. 367-377 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Normal Family ◽  
Entire Functions ◽  
Julia Set ◽  
Maximum Modulus ◽  
Stable Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Simply Connected

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

scholarly journals Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

2021 ◽  
Author(s):  
D. A. Nicks ◽  
P. J. Rippon ◽  
G. M. Stallard
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Regular Growth ◽  
Transcendental Entire Function ◽  
Minimum Modulus ◽  
Precise Control ◽  
Significant Rate ◽  
Small Growth ◽  
Minimal Type

AbstractFor a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ m ( r ) = min { | f ( z ) | : | z | = r } , is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg’s method of constructing entire functions of small growth, which allows rather precise control of m(r).

Interpolation in Spaces of Entire Functions in ℂN

1976 ◽  
Vol 19 (1) ◽  
pp. 109-112 ◽  
Author(s):  
Lawrence Gruman
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Growth Conditions ◽  
Set Of Points

Let {sm} be a discrete set of points in ℂN and λm any sequence of points in ℂ. We shall be interested in finding an entire function F(z) such that F(sm)=λm. This is of course easy if no restriction is placed on F, but we shall be interested in finding an F which in addition satisfies certain growth conditions.We shall denote the variable z=(z1,..., zN), zj=xj+iyj,

scholarly journals Entire Functions of Bounded L-Index: Its Zeros and Behavior of Partial Logarithmic Derivatives

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Maximum Modulus ◽  
Minimum Modulus ◽  
Distribution Of Zeros ◽  
And Behavior ◽  
Logarithmic Derivatives

In this paper, we obtain new sufficient conditions of boundedness of L-index in joint variables for entire function in Cn functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and l-index in C too. They generalize known results of Fricke, Sheremeta, and Kuzyk.

On the Growth of Entire Function Defined by Multiple Taylor

2014 ◽  
Vol 905 ◽  
pp. 786-789
Author(s):  
Wan Chun Lu ◽  
You Hua Peng
Keyword(s):  
Entire Function ◽  
Taylor Series ◽  
Necessary Conditions ◽  
Entire Functions ◽  
Polar Coordinates ◽  
Growth Of Entire Functions ◽  
Sufficient And Necessary Conditions

It is studied that the growth of entire functions defined by multiple Taylor series by means of Polar coordinates, and established two sufficient and necessary conditions.

scholarly journals Connectedness properties of the set where the iterates of an entire function are bounded

2013 ◽  
Vol 155 (3) ◽  
pp. 391-410 ◽  
Author(s):  
JOHN OSBORNE
Keyword(s):  
Entire Function ◽  
Critical Point ◽  
Entire Functions ◽  
Empty Interior ◽  
Transcendental Entire Functions ◽  
Set Of Points ◽  
Quasiconformal Surgery ◽  
Totally Disconnected

AbstractWe investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.

scholarly journals Uniqueness on entire functions and their nth order exact differences with two shared values

2020 ◽  
Vol 18 (1) ◽  
pp. 211-215
Author(s):  
Shengjiang Chen ◽  
Aizhu Xu
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Shared Values ◽  
Simple Method ◽  
Exact Difference ◽  
Hyper Order

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

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