Hyper Relative Order () of Entire Functions

doi:10.1515/AWUTM-2017-0015

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Hyper Relative Order () of Entire Functions

Journal Article•DOI•

Hyper Relative Order () of Entire Functions

Dibyendu Banerjee¹, Saikat Batabyal•Institutions (1)

01 Dec 2017-Annals of the West University of Timisoara: Mathematics and Computer Science (Walter de Gruyter GmbH)-Vol. 55, Iss: 2, pp 65-84

TL;DR: In this article, the authors introduced hyper relative order (p, q) of entire functions where p, q are positive integers with p>q and proved sum theorem, product theorem and theorem on derivative.

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Abstract: Abstract After 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 where p, q are positive integers with p>q and prove sum theorem, product theorem and theorem on derivative.

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Book•

The theory of functions

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E. C. Titchmarsh

01 Jan 1932

TL;DR: Alfaro et al. as mentioned in this paper conservado en la Biblioteca del Campus de Mostoles de la Universidad Rey Juan Carlos (sign. 517.5 TIT THE).

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Abstract: Original conservado en la Biblioteca del Campus de Mostoles de la Universidad Rey Juan Carlos (sign. 517.5 TIT THE).

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2,695 citations

"Hyper Relative Order () of Entire F..." refers background in this paper

...When g(z) = exp(z), ρg(f) coincides with the classical definition of order ([15],p-248)....
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Monograph•DOI•

Distribution of zeros of entire functions

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B. I︠a︡. Levin

31 Dec 1964

1,788 citations

"Hyper Relative Order () of Entire F..." refers background in this paper

...([12], p-21) Let f(z) be holomorphic in the circle |z| = 2eR(R > 0) with f(0) = 1 and η be an arbitrary positive number not exceeding 3e 2 ....
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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.

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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

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101 citations

"Hyper Relative Order () of Entire F..." refers background in this paper

...[14] D. Sato, On the rate of growth of entire functions of fast growth, Bull....
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...Following Sato [14], we write log[0] x = x, exp[0] x = x and for positive integer m ≥ 1, log[m] x = log(log[m−1] x), exp[m] x = exp(exp[m−1] x)....
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...Following Sato [14], we write log x = x, exp x = x and for positive integer m ≥ 1, log x = log(log[m−1] x), exp x = exp(exp[m−1] x)....
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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.

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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.

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69 citations

"Hyper Relative Order () of Entire F..." refers background in this paper

...In 1988, Bernal [2] introduced the definition of relative order of f with respect to g as ρg(f) = inf{μ > 0 : Mf (r) < Mg(r) for all r > r0(μ) > 0}....
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...[2] Let g be an entire function which satisfies the property (A), and let σ > 1....
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...In 1988, Bernal [2] introduced the definition of relative order of f with respect to g as ρg(f) = inf{µ > 0 : Mf (r) < Mg(rµ) for all r > r0(µ) > 0}....
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...The following definition of Bernal [2] will be needed....
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...[2] Suppose f is an entire function, α > 1, 0 < β < α, s > 1, 0 < μ < λ and n is a positive integer....
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Journal Article•

On the (p, q)-type and lower (p, q)-type of an entire function.

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S. K. Bajpai, O. P. Juneja, G. P. Kapoor

01 Jan 1977-Crelle's Journal

TL;DR: In this article, the authors introduced the concepts of (p, #)-type and lower (/?, #) type for the function /(z) and obtained their characterizations in terms of the coefficients an in the Taylor series (1. 1).

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Abstract: be a nonconstant entire fimction where A0 = 0 and {An}*=1 is the strictly increasing sequence of positive integers such that an Φ 0 for n = l, 2, 3, . . . . The rate of growth of/(z) is studied in [8] in a general manner in terms of its (/?, #)-order and lower (p,

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62 citations