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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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


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

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

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40E05, 26A12

Retrieve articles in all journals with MSC: 40E05, 26A12

Additional Information

Keywords: Abel-Tauber theorems, regular variation
Article copyright: © Copyright 1981 American Mathematical Society