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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ \Pi $-regular variation


Author: J. L. Geluk
Journal: Proc. Amer. Math. Soc. 82 (1981), 565-570
MSC: Primary 40E05; Secondary 26A12
MathSciNet review: 614879
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Abstract: A function $ U:{R^ + } \to {R^ + }$ is said to be $ \Pi $-regularly varying with exponent $ \alpha $ if $ U(x){x^{ - \alpha }}$ is nondecreasing and there exists a positive function $ L$ such that

$\displaystyle \frac{{U(\lambda x)/{\lambda ^\alpha } - U(x)}} {{{x^\alpha }L(x)... ...o \log \lambda \quad (x \to \infty ){\text{for}}\lambda {\text{ > 0}}{\text{.}}$

Suppose

$\displaystyle \hat{U}(t)=\int _0^\infty {e^{ - tx}}dU(x){\text{exists for }}t > 0.$

We prove that $ U$ is $ \Pi $-regularly varying iff $ U$ is $ \Pi $-regularly varying.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614879-X
PII: S 0002-9939(1981)0614879-X
Keywords: Abel-Tauber theorems, regular variation
Article copyright: © Copyright 1981 American Mathematical Society