Derivation of Some Results on the Generalized Relative Orders of Meromorphic Functions

doi:10.1515/AWUTM-2017-0004

Home
/
Papers
/
Derivation of Some Results on the Generalized Relative Orders of Meromorphic Functions

Journal Article•DOI•

Derivation of Some Results on the Generalized Relative Orders of Meromorphic Functions

Sanjib Kumar Datta¹, Tanmay Biswas•Institutions (1)

26 Jul 2017-Annals of the West University of Timisoara: Mathematics and Computer Science (Sciendo)-Vol. 55, Iss: 1, pp 51-61

TL;DR: In this paper, the relative lower order of a meromorphic function f with respect to another entire function g was investigated when generalized relative order (generalized relative lower orders) of f and generalized relative ordering of g were given.

read less

Abstract: Abstract In this paper we intend to find out relative order (relative lower order) of a meromorphic function f with respect to another entire function g when generalized relative order (generalized relative lower order) of f and generalized relative order (generalized relative lower order) of g with respect to another entire function h are given.

...read moreread less

Content maybe subject to copyright Report

Citations

PDF

Open Access

More filters

Journal Article•

Growth of entire functions of two complex variable based on relative order

[...]

Bal Ram Prajapati, Anupma Rastogi

23 Dec 2015-International Journal of Mathematical Archive

TL;DR: In this paper, the authors introduce the idea of comparative growth properties of entire functions of two complex variables on the basis of relative order, relative order and relative -order and introduce the concept of relative-order growth.

...read moreread less

Abstract: I n this paper we introduce the idea of comparative growth properties of entire functions of two complex variable on the basis relative order, relative -order and relative -order

...read moreread less

9 citations

References

PDF

Open Access

More filters

Journal Article•DOI•

On the characteristic of functions meromorphic in the unit disk and of their integrals

[...]

W. K. Hayman

01 Dec 1964-Acta Mathematica

TL;DR: In this paper, the Nevanlinna characteristic function of a regular function is defined as a convex increasing function with bounded characteristic in the sense that it always exists as a finite or infinite limit.

...read moreread less

Abstract: n(r, F) as the number of poles in ]z] d r and = f r n(t, F) dt N(r, F) J0 $ Then T(r, F) = re(r, F) + N(r, F) is called the Nevanlinna characteristic function of F(z). The function T(r, F ) i s convex increasing function of log r, so tha t T(1, F) = lim T(r, F) r--~l always exists as a finite or infinite limit. I f T(1, F) is finite we say tha t F(z) has bounded characteristic in ]z] < 1. Examples show tha t F(z) m a y have bounded characteristic in ]z] < 1, even i f / (z) does not.(1) We may take for instance /(z) to be a regular function

...read moreread less

614 citations

"Derivation of Some Results on the G..." refers background in this paper

...We do not explain the standard definitions and notations in the theory of entire and meromorphic functions as those are available in [11] and [14]....
[...]

Book•

Lectures on the general theory of integral functions

[...]

Georges Valiron

01 Jan 1923

489 citations

"Derivation of Some Results on the G..." refers background in this paper

...We do not explain the standard definitions and notations in the theory of entire and meromorphic functions as those are available in [11] and [14]....
[...]

Journal Article•DOI•

On the rate of growth of entire functions of fast growth

[...]

Daihachiro Sato

01 May 1963-Bulletin of the American Mathematical Society

TL;DR: In this article, the authors define lemmas as functions expx, log#, E[r]-x, AM(X), E^ ^) (m = 0, ± 1, ± 2, • • • ; r= 0, 1, 2,• • • ) all increase monotonically.

...read moreread less

Abstract: 2. Definitions. Notations and preparatory lemmas. NOTATION 1. exp =log ^ = x, exp# = log~x = exp(exp'-%) = log(logt~-%) (m = 0, ± 1 , ± 2 , • • • ). NOTATION 2. r r £«(*) n «pro *, Aw(«) n iog w *, E[-r](x) = »/A[r-i](»), Afr-riO*) = a?/£[r-.i](«), x = £M(y) « y = £[r](«) (f = 0, ± 1, ± 2 , • • • ). LEMMAS. 77^ functions expx, log#, E[r](x), AM(X), E^ ^) (m = 0, ± 1 , ± 2 , • • • ; r = 0, 1, 2, • • • ) all increase monotonically and we have

...read moreread less

101 citations

"Derivation of Some Results on the G..." refers methods in this paper

...In the sequel the following two notations are used: log[k] x = log ( log[k−1] x ) for k = 1, 2, 3, · · · ; log[0] x = x and exp[k] x = exp ( exp[k−1] x ) for k = 1, 2, 3, · · · ; exp[0] x = x. Taking this into account the generalized order (respectively, generalized lower order) of an entire function f as introduced by Sato [13] is given by: ρ [l] f = lim sup r→∞ log[l]Mf (r) log logMexp z (r) = lim sup r→∞ log[l]Mf (r) log r( respectively λ [l] f = lim infr→∞ log[l]Mf (r) log logMexp z (r) = lim inf r→∞ log[l]Mf (r) log r ) where l ≥ 1....
[...]
...Taking this into account the generalized order (respectively, generalized lower order) of an entire function f as introduced by Sato [13] is given by:...
[...]
...[13] D. Sato, On the rate of growth of entire functions of fast growth, Bull....
[...]

Journal Article•

Orden relativo de crecimiento de funciones enteras

[...]

Luis Bernal González

01 Jan 1988-Collectanea Mathematica

TL;DR: In this paper, a generalization of the classical concept of growth order of an entire function is proposed, where the authors define the new parameter ρ_g(f), the relative growth order with respect to ρ(z), and establish a direct comparison between the growth of the moduli of two nonconstant entire functions.

...read moreread less

Abstract: In this paper, we essay a generalization of the classical concept of growth order of an entire function. We define the new parameter $\rho_g(f)$, the relative growth order of $f(z)$ with respect to $g(z)$, which establishes a direct comparison between the growth of the moduli of two nonconstant entire functions $f$ and $g$. Diverse properties, relative to sum, product, composition, derivative, real and imaginary parts, Nevanlinna’s characteristic and Taylor’s coefficients are studied.

...read moreread less

69 citations

"Derivation of Some Results on the G..." refers background in this paper

...Extending the idea of relative order of entire functions as established by Bernal {[1], [2]} , Lahiri and Banerjee [12] introduced the definition of relative order of a meromorphic function f with respect to another entire function g, denoted by ρg (f) to avoid comparing growth just with exp z as follows:...
[...]
...These definitions extend the definitions of order ρf and lower order λf of an entire or meromorphic function f since for l = 2, these correspond to the particular case ρ [2] f = ρf and λ [2] f = λf ....
[...]

Measure of growth ratios of composite entire and meromorphic functions with a focus on relative order

[...]

Sanjib Kumar Datta, Tanmay Biswas, Chinmay Biswas

01 Jan 2014

TL;DR: In this paper, comparative growth properties of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders are discussed, where the authors show that composite entire functions have better growth properties than their meromorphic counterparts.

...read moreread less

Abstract: Some comparative growth properties of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders are discussed in this paper.

...read moreread less

19 citations

"Derivation of Some Results on the G..." refers background in this paper

...Therefore the growth of composite entire and meromorphic functions needs to be modified on the basis of their relative order some of which has been explored in [4], [5], [6], [7], [8], [9] and [10]....
[...]