scholarly journals Hyper Relative Order (p, q) of Entire Functions

2017 ◽  
Vol 55 (2) ◽  
pp. 65-84
Author(s):  
Dibyendu Banerjee ◽  
Saikat Batabyal
Keyword(s):  
Entire Functions ◽  
Relative Order ◽  
Product Theorem ◽  
Sum Theorem ◽  
Positive Integers

AbstractAfter the works of Lahiri and Banerjee [6] on the idea of relative order (p, q) of entire functions, we introduce in this paper hyper relative order (p, q) of entire functions wherep, qare positive integers withp>qand prove sum theorem, product theorem and theorem on derivative.

scholarly journals On the(p,q)th Relative Order Oriented Growth Properties of Entire Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Manuel Sánchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal
Keyword(s):  
Quantitative Assessment ◽  
Entire Functions ◽  
Relative Order ◽  
Order Of Growth ◽  
Alternative Definition ◽  
Oriented Growth ◽  
Growth Properties ◽  
Self Similar ◽  
Definition Of ◽  
Positive Integers

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integerspandq, we wish to introduce an alternative definition of relative(p,q)th order which improves the earlier definition of relative(p,q)th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative(p,q)th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

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 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.

Some growth properties of composite p-adic entire functions on the basis of their relative order and relative lower order

2019 ◽  
Vol 12 (03) ◽  
pp. 1950044
Author(s):  
Tanmay Biswas
Keyword(s):  
Growth Rates ◽  
Lower Order ◽  
Entire Functions ◽  
Closed Field ◽  
Relative Order ◽  
Algebraically Closed Field ◽  
Growth Properties

Let [Formula: see text] be a complete ultrametric algebraically closed field and [Formula: see text] be the [Formula: see text]-algebra of entire functions on [Formula: see text]. For [Formula: see text], [Formula: see text], we wish to introduce the notions of relative order and relative lower order of [Formula: see text] with respect to [Formula: see text]. Hence, after proving some basic results, in this paper, we estimate some growth rates of composite p-adic entire functions on the basis of their relative orders and relative lower orders.

scholarly journals Growth Estimates of Composite Entire Functions in the Light of Slowly Changing Functions Based Relative Order, Relative Type and Relative Weak Type

2015 ◽  
Vol 54 (1) ◽  
pp. 59-74
Author(s):  
S. K. Datta ◽  
T. Biswas ◽  
S. Bhattacharyya
Keyword(s):  
Weak Type ◽  
Entire Functions ◽  
Maximum Modulus ◽  
Relative Order ◽  
Maximum Term ◽  
Growth Properties ◽  
Relative Type

Abstract In the paper we prove some growth properties of maximum term and maximum modulus of composition of entire functions on the basis of relative L*-order, relative L*-type and relative L*-weak type.

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.

Restrictions on meromorphic solutions of Fermat type equations

2020 ◽  
Vol 63 (3) ◽  
pp. 654-665
Author(s):  
Gary G. Gundersen ◽  
Katsuya Ishizaki ◽  
Naofumi Kimura
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Existence Theorems ◽  
Rational Functions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Nevanlinna Functions ◽  
Positive Integers

AbstractThe Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$, where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

Relative (p,q)-φ order and relative (p,q)-φ type based on some growth properties of composite p-adic entire functions

2020 ◽  
Vol 29 (1) ◽  
pp. 09-16
Author(s):  
Biswas Tanmay
Keyword(s):  
Weak Type ◽  
Entire Functions ◽  
Closed Field ◽  
Unbounded Function ◽  
Algebraically Closed Field ◽  
Growth Properties ◽  
Positive Integers

Let K be a complete ultrametric algebraically closed field and A (K) be the K-algebra of entire functions on K. For any p adic entire functions f ∈ A (K) and r > 0, we denote by |f| (r) the number sup {|f (x) | : |x| = r} where |·| (r) is a multiplicative norm on A (K) . In this paper we study some growth properties of composite p-adic entire functions on the basis of their relative (p, q)-ϕ order, relative (p, q)-ϕ type and relative (p, q)-ϕ weak type where p, q are any two positive integers and ϕ (r) : [0, +∞) → (0, +∞) is a non-decreasing unbounded function of r.

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