On the order of functions at infinity

Cadena, Meitner; Kratz, Marie; Omey, Edward

doi:10.1016/j.jmaa.2017.02.042

Cited by 8 publications

(15 citation statements)

References 17 publications

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“…The result follows since lim (ii) Using the representation theorem for M(ρ) (see [8], Theorem 1.2), we have…”

Section: Some Propertiesmentioning

confidence: 83%

“…Proof of Theorem 2.6. The proof of the theorem is based on the following lemma, inspired by Theorem 1.1 in [5] or [8].…”

Section: Characterization Theoremmentioning

confidence: 99%

“…Assume now the conditions given in [8], Lemma 1.2, namely that U is a nondecreasing right continuous function with support R + and U (0+) = 0, that U (s) < ∞, ∀s > 0, and that x −η U (x) is a concave function for some real number η > 0. Then we have the following Karamata Tauberian type of theorem.…”

Section: Laplace Transformsmentioning

confidence: 99%

“…First suppose that U ∈ M 0 (L, ρ). Using (9) with V defined in 8, and Lemma 1.2 in [8], we can say that there exist positive constants a, b, c, such that, ∀x > x ǫ ,…”

Section: Laplace Transformsmentioning

confidence: 99%

“…It has already been proved in [8] that U ∈ RV ρ ⇒ U ∈ M(ρ) but that the converse is false. We also considered there O− type of relations and, among others, studied functions for which lim sup x→∞ log U (x)/ log x < ∞.…”

Section: Introductionmentioning

confidence: 99%

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New Results on the Order of Functions at Infinity

Cadena¹,

Kratz²,

Omey³

2017

SSRN Journal

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Recently, new classes of positive and measurable functions, M(ρ) and M(±∞), have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., , 2017. Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type lim x→∞ log U (x)/H(x) = ρ < ∞ for a large class of normalizing functions H. It provides subclasses of M(0) and M(±∞).

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“…The result follows since lim (ii) Using the representation theorem for M(ρ) (see [8], Theorem 1.2), we have…”

Section: Some Propertiesmentioning

confidence: 83%

“…Proof of Theorem 2.6. The proof of the theorem is based on the following lemma, inspired by Theorem 1.1 in [5] or [8].…”

Section: Characterization Theoremmentioning

confidence: 99%

Section: Laplace Transformsmentioning

confidence: 99%

“…First suppose that U ∈ M 0 (L, ρ). Using (9) with V defined in 8, and Lemma 1.2 in [8], we can say that there exist positive constants a, b, c, such that, ∀x > x ǫ ,…”

Section: Laplace Transformsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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