scholarly journals On entire functions with gap power series

1971 ◽  
Vol 12 (2) ◽  
pp. 89-97 ◽  
Author(s):  
J. M. Anderson ◽  
K. G. Binmore
Keyword(s):  
Power Series ◽  
Lower Order ◽  
Entire Functions ◽  
Transcendental Entire Functions ◽  
Image Position ◽  
Positive Integers

In this note we consider transcendental entire functionswhose power series contain gaps, i.e.where Λ = {λk} is a suitable set of positive integers. We denote the set of all such functions f(z) by E(Λ). As usual M(r) = M(r, f) denotes the maximummodulus of f(z) on the circle |z| = r. The order p and the lower order λ of f(z) are defined byrespectively.

Entire Functions with Some Derivatives Univalent

1974 ◽  
Vol 26 (1) ◽  
pp. 207-213 ◽  
Author(s):  
S. M. Shah ◽  
S. Y. Trimble
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Radius Of Convergence ◽  
Open Disc ◽  
Image Position ◽  
Positive Integers ◽  
The Relationship ◽  
Derivatives Of

This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(np) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals On relative (p,q)-th order based growth measure of entire functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1723-1735
Author(s):  
Sanjib Datta ◽  
Tanmay Biswas ◽  
Chinmay Ghosh
Keyword(s):  
Entire Function ◽  
Lower Order ◽  
Entire Functions ◽  
Positive Integers

In this paper we intend to find out relative (p,q)-th order (relative (p,q)-th lower order) of an entire function f with respect to another entire function 1 when relative (m,q)-th order (relative (m,q)-th lower order) of f and relative (m,p)-th order (relative (m,p)-th lower order) of g with respect to another entire function h are given with p,q,m are all positive integers.

INTERPRETING ARITHMETIC IN THE FIRST-ORDER THEORY OF ADDITION AND COPRIMALITY OF POLYNOMIAL RINGS

2019 ◽  
Vol 84 (3) ◽  
pp. 1194-1214
Author(s):  
JAVIER UTRERAS
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Order Theory ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Series Ring ◽  
First Order ◽  
First Order Theory ◽  
Power Series Rings ◽  
Image Position

AbstractWe study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, + , \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself.

scholarly journals $(p,q)$th order oriented growth measurement of composite $p$-adic entire functions

2018 ◽  
Vol 10 (2) ◽  
pp. 248-272
Author(s):  
T. Biswas
Keyword(s):  
Lower Order ◽  
Complex Analysis ◽  
Entire Functions ◽  
Log P ◽  
Closed Field ◽  
Order Of Growth ◽  
Oriented Growth ◽  
Growth Measurement ◽  
Growth Properties ◽  
Positive Integers

Let $\mathbb{K}$ be a complete ultrametric algebraically closed field and let $\mathcal{A}\left(\mathbb{K}\right)$ be the $\mathbb{K}$-algebra of entire functions on $\mathbb{K}$. For any $p$-adic entire function $f\in \mathcal{A}\left( \mathbb{K}\right) $ and $r>0$, we denote by $|f|\left(r\right)$ the number $\sup \left\{ |f\left( x\right) |:|x|=r\right\}$, where $\left\vert \cdot \right\vert (r)$ is a multiplicative norm on $\mathcal{A}\left( \mathbb{K}\right)$. For any two entire functions $f\in \mathcal{A}\left(\mathbb{K}\right)$ and $g\in \mathcal{A}\left(\mathbb{K}\right)$ the ratio $\frac{|f|(r)}{|g|(r)}$ as $r\rightarrow \infty $ is called the comparative growth of $f$ with respect to $g$ in terms of their multiplicative norms. Likewise to complex analysis, in this paper we define the concept of $(p,q)$th order (respectively $(p,q)$th lower order) of growth as $\rho ^{\left( p,q\right) }\left( f\right) =\underset{r\rightarrow +\infty }{\lim \sup } \frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$ (respectively $\lambda ^{\left( p,q\right) }\left( f\right) =\underset{ r\rightarrow +\infty }{\lim \inf }\frac{\log ^{[p]}|f|\left( r\right) }{\log ^{\left[ q\right] }r}$), where $p$ and $q$ are any two positive integers. We study some growth properties of composite $p$-adic entire functions on the basis of their $\left(p,q\right)$th order and $(p,q)$th lower order.

On the iteration of rational functions

1978 ◽  
Vol 84 (3) ◽  
pp. 497-505 ◽  
Author(s):  
V. Garber
Keyword(s):  
Rational Function ◽  
Entire Functions ◽  
Rational Functions ◽  
Transcendental Entire Function ◽  
The Family ◽  
Transcendental Entire Functions ◽  
Definition Of ◽  
Iteration Of Rational Functions ◽  
Image Position ◽  
Set Of Points

In the theory of the iteration of a rational function or transcendental entire function R(z) of the complex variable z we study the sequence of natural iterates, {Rn(z):n = 0, 1,…}, of R, whereThe domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F-set, F(R), of R, R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.

scholarly journals A family of entire functions with Baker domains

2009 ◽  
Vol 29 (2) ◽  
pp. 495-514
Author(s):  
DOMINIQUE S. FLEISCHMANN
Keyword(s):  
Entire Functions ◽  
Transcendental Entire Functions ◽  
Novel Method ◽  
Image Position

AbstractIn his paper [The iteration of polynomials and transcendental entire functions. J. Aust. Math. Soc. (Series A)30 (1981), 483–495], Baker proved that the function f defined by has a Baker domain for c sufficiently large. In this paper we use a novel method to prove that f has a Baker domain for all c>0. We also prove that there exists an open unbounded set contained in the Baker domain on which the orbits of points under f are asymptotically horizontal.

scholarly journals An elementary existence theorem for entire functions

1971 ◽  
Vol 5 (3) ◽  
pp. 415-419 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Existence Theorem ◽  
Entire Functions ◽  
Real Numbers ◽  
Transcendental Entire Functions ◽  
Image Position

It is proved that, for any m given distinct real numbers a1, …, am, there exist transcendental entire functions f(z) at most of order m for which all the valuesare rational integers.

scholarly journals On Some Growth Properties of Entire Functions Using Their Maximum Moduli Focusingp,qth Relative Order

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Luis Manuel Sanchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Entire Function ◽  
Growth Rates ◽  
Lower Order ◽  
Entire Functions ◽  
Relative Order ◽  
Growth Properties ◽  
Definition Of ◽  
Positive Integers

We discuss some growth rates of composite entire functions on the basis of the definition of relativep,qth order (relativep,qth lower order) with respect to another entire function which improve some earlier results of Roy (2010) wherepandqare any two positive integers.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

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