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Real inequalities with applications to function theory

Published online by Cambridge University Press:  24 October 2008

W. K. Hayman  and
F. M. Stewart
W. K. Hayman
Affiliation:
University CollegeExeter
F. M. Stewart
Affiliation:
Brown UniversityProvidence, R.I.
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Extract

Suppose that f(x) is non-negative for x ≥ 0 and let

In a recent paper (4) inequalities were proved relating fn(x) with the nth derivative f(n)(x) of f(x). However, the two main results, Theorems 3 and 4, were proved only for n = 1 and 2. In the first part of this paper we shall prove the corresponding results for all positive integral n under a slightly less restrictive hypothesis (Theorem 3). We shall then give an application showing that from the hypothesis f(x) ≤ μ(x) for all x it is sometimes possible to deduce f(n)(x) = O{μ(n)(x)} for a set of values x depending only on μ(x).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 250 - 260
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

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