### Abstract

A real-valued function f of a real variable is said to be φ{symbol}-slowly varying (φ{symbol}-s.v.) if lim_{x→∞}φ{symbol}(x) [f(x+α)-f(x)]=0 for each α. It is said to be uniformly φ{symbol}-slowly varying (u. φ{symbol}-s.v.) if lim_{x→∞} sup_{α ∈ I}φ{symbol}(x) |f(x+α)-f(x)|=0 for every bounded interval I. It is supposed throughout that φ{symbol} is positive and increasing. It is proved that if φ{symbol} increases rapidly enough, then every φ{symbol}-s.v. function f must be u. φ{symbol}-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of φ{symbol}, a measurable function f must be u. φ{symbol}-s.v. if it is φ{symbol}-s.v. Examples of pairs (φ{symbol},f) are given that illustrate the necessity for the requirements on φ{symbol} and f in these results.

Original language | English |
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Pages (from-to) | 1-9 |

Number of pages | 9 |

Journal | Aequationes Mathematicae |

Volume | 10 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1974 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Aequationes Mathematicae*,

*10*(1), 1-9. https://doi.org/10.1007/BF01834775

**Very slowly varying functions.** / Ash, J. Marshall; Erdős, P.; Rubel, L. A.

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 10, no. 1, pp. 1-9. https://doi.org/10.1007/BF01834775

}

TY - JOUR

T1 - Very slowly varying functions

AU - Ash, J. Marshall

AU - Erdős, P.

AU - Rubel, L. A.

PY - 1974/2

Y1 - 1974/2

N2 - A real-valued function f of a real variable is said to be φ{symbol}-slowly varying (φ{symbol}-s.v.) if limx→∞φ{symbol}(x) [f(x+α)-f(x)]=0 for each α. It is said to be uniformly φ{symbol}-slowly varying (u. φ{symbol}-s.v.) if limx→∞ supα ∈ Iφ{symbol}(x) |f(x+α)-f(x)|=0 for every bounded interval I. It is supposed throughout that φ{symbol} is positive and increasing. It is proved that if φ{symbol} increases rapidly enough, then every φ{symbol}-s.v. function f must be u. φ{symbol}-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of φ{symbol}, a measurable function f must be u. φ{symbol}-s.v. if it is φ{symbol}-s.v. Examples of pairs (φ{symbol},f) are given that illustrate the necessity for the requirements on φ{symbol} and f in these results.

AB - A real-valued function f of a real variable is said to be φ{symbol}-slowly varying (φ{symbol}-s.v.) if limx→∞φ{symbol}(x) [f(x+α)-f(x)]=0 for each α. It is said to be uniformly φ{symbol}-slowly varying (u. φ{symbol}-s.v.) if limx→∞ supα ∈ Iφ{symbol}(x) |f(x+α)-f(x)|=0 for every bounded interval I. It is supposed throughout that φ{symbol} is positive and increasing. It is proved that if φ{symbol} increases rapidly enough, then every φ{symbol}-s.v. function f must be u. φ{symbol}-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of φ{symbol}, a measurable function f must be u. φ{symbol}-s.v. if it is φ{symbol}-s.v. Examples of pairs (φ{symbol},f) are given that illustrate the necessity for the requirements on φ{symbol} and f in these results.

UR - http://www.scopus.com/inward/record.url?scp=33745840640&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745840640&partnerID=8YFLogxK

U2 - 10.1007/BF01834775

DO - 10.1007/BF01834775

M3 - Article

AN - SCOPUS:33745840640

VL - 10

SP - 1

EP - 9

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 1

ER -