Some properties of asymptotic functions and their applications

Chan, Ling Yau

doi:10.1090/S0002-9939-1978-0507315-5

Some properties of asymptotic functions and their applications
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Proc. Amer. Math. Soc. 72 (1978), 239-247 Request permission

Abstract:

In this paper we give complete characterizations, in terms of Dini numbers and integrals, of positive functions $\Phi (u)$ defined in (0, $\infty$) satisfying the conditions: (i) $\Phi (u)/{u^a}$ is nondecreasing and (ii) $\Phi (u)/{u^b}$ is nonincreasing. By applying these results we obtain necessary and sufficient conditions for power series and trigonometric series to satisfy a certain Lipschitz condition, which include some known results of R. P. Boas, Jr. [1]. We also give complete characterizations of positive functions $\Phi (u)$ defined in $( - \infty ,\infty )$ satisfying the conditions: (i) $\Phi (u)/{e^{au}}$ is nondecreasing and (ii) $\Phi (u)/{e^{bu}}$ is nonincreasing.

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