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.

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 A Note on Relative $(p,q)$ th Proximate Order of Entire Functions

2016 ◽  
Vol 8 (5) ◽  
pp. 1
Author(s):  
Luis Manuel Sanchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Chinmay Ghosh
Keyword(s):  
Entire Function ◽  
Physical State ◽  
Entire Functions ◽  
Relative Order ◽  
Proximate Order ◽  
Positive Integers

Relative order of functions measures specifically how different in growth two given functions are which helps to settle the exact physical state of a system. In this paper for any two positive integers $p$ and $q,$ we introduce the notion of relative $(p,q)$ th proximate order of an entire function with respect to another entire function and prove its existence.

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.

scholarly journals Fast escaping points of entire functions: a new regularity condition

2015 ◽  
Vol 160 (1) ◽  
pp. 95-106
Author(s):  
V. EVDORIDOU
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Lower Order ◽  
Regularity Condition ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Transcendental Dynamics

AbstractLet f be a transcendental entire function. The fast escaping set, A(f), plays a key role in transcendental dynamics. The quite fast escaping set, Q(f), defined by an apparently weaker condition is equal to A(f) under certain conditions. Here we introduce Q2(f) defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy Q2(f) = A(f). We also show that the finite composition of such functions satisfies Q2(f) = A(f). Finally, we construct a function for which Q2(f) ≠ Q(f) = A(f).

Growth Analysis of Composite Entire Functions of Several Complex Variables Based on Relative Order

2020 ◽  
pp. 151-163
Author(s):  
Balram Prajapati ◽  
Anupama Rastogi
Keyword(s):  
Entire Function ◽  
Lower Order ◽  
Growth Analysis ◽  
Entire Functions ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Relative Order ◽  
Growth Properties

<p>In this paper we introduce some new results depending on the comparative growth properties of composition of entire function of several complex variables using relative L^*-order, Relative L^*-lower order and L≡L(r_1,r_2,r_3,……..,r_n) is a slowly changing functions. We prove some relation between relative L^*- order and relative L^*- lower order.</p>

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.

scholarly journals Entire functions having asymptotic functions

1983 ◽  
Vol 27 (3) ◽  
pp. 321-328 ◽  
Author(s):  
P.C. Fenton
Keyword(s):  
Entire Function ◽  
Lower Order ◽  
Entire Functions ◽  
Asymptotic Functions

It is shown that an entire function having k distinct entire asymptotic functions of order less than ¼ is of lower order ½k, mean type at least; further that if f is of lower order ½k, mean type, then its order is ½k.

Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones

2019 ◽  
Vol 63 (3) ◽  
pp. 536-546
Author(s):  
Taboka Prince Chalebgwa
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Lower Order ◽  
Entire Functions ◽  
Compact Sets ◽  
Bounded Degree ◽  
Growth Of Entire Functions ◽  
Original Theorem

AbstractGiven an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order $\unicode[STIX]{x1D706}$, Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of $f$. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply.

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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