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 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 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).

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 Dynamical sets whose union with infinity is connected

2018 ◽  
Vol 40 (3) ◽  
pp. 789-798 ◽  
Author(s):  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Large Class ◽  
Topological Structure ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Related Question ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Class Of Functions

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.

scholarly journals On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients

2021 ◽  
Vol 56 (2) ◽  
pp. 149-161
Author(s):  
T. H. Nguyen ◽  
A. Vishnyakova
Keyword(s):  
Entire Function ◽  
Theta Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Taylor Coefficients ◽  
Partial Theta Function ◽  
Necessary And Sufficient

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.

scholarly journals Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 88 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv
Keyword(s):  
Continuous Function ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Maximum Modulus ◽  
Quaternionic Analysis ◽  
Directional Derivatives ◽  
Positive Continuous Function ◽  
Several Variables ◽  
Necessary And Sufficient ◽  
Class Of Functions

In this paper, for a given direction b ∈ C n \ { 0 } we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line { z 0 + t b : t ∈ C } for any z 0 ∈ C n . Unlike to quaternionic analysis, we fix the direction b . The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z 1 and continuous in variable z 2 . For this class of functions there is introduced a concept of boundedness of L-index in the direction b where L : C n → R + is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function L : C n → R + .

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.

scholarly journals The minimum modulus of gap power series

1978 ◽  
Vol 21 (1) ◽  
pp. 49-54 ◽  
Author(s):  
P. C. Fenton
Keyword(s):  
Entire Function ◽  
Power Series ◽  
Maximum Modulus ◽  
Minimum Modulus ◽  
Image Position

Letbe an entire function, where (αn) is a strictly increasing sequence of non-negativeintegers. The maximum modulus, M(r), the minimum modulus, m(r), and the maxi-mum term, μ(r), of f defined by

scholarly journals Approximation of entire functions over Carathéodory domains

1982 ◽  
Vol 25 (2) ◽  
pp. 221-229 ◽  
Author(s):  
G.P. Kapoor ◽  
A. Nautiyal
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Finite Type ◽  
Finite Order ◽  
Jordan Curve ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Image Position ◽  
Necessary And Sufficient

Let D be a domain bounded by a Jordan curve. For 1 ≤ p ≤ ∞, let Lp(D) be the class of all functions f holomorphic in D such that where A is the area of D. For f ∈Lp(D), setπn consists of all polynomials of degree at most n. Recently, Andre Giroux (J. Approx. Theory 28 (1980), 45–53) has obtained necessary and sufficient conditions, in terms of the rate of decrease of the approximation error , such that has an analytic continuation as an entire function having finite order and finite type. In the present paper we have considered the approximation error (*) on a Carathéodory domain and have extended the results of Giroux for the case 1 ≤ p < 2.

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