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

 

A modulus of continuity for a class of quasismooth functions


Author: J. Ernest Wilkins
Journal: Proc. Amer. Math. Soc. 93 (1985), 459-465
MSC: Primary 26A15; Secondary 26D20
MathSciNet review: 774003
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Abstract: Let $ Z$ be the class of real-valued functions $ f(x)$, defined and continuous on the closed interval $ I = [ - 1,1]$, such that $ f( - 1) = f(1) = 0$ and

$\displaystyle \vert f(\xi ) - 2f\{ (\xi + \eta )/2\} + f(\eta )\vert \leqslant \vert\xi - \eta \vert$

for all $ \xi $ and $ \eta $ in $ I$. We show that $ \omega (h) = h{\log _2}\left\{ {2eK/(h{{\log }_2}e)} \right\}$ is a modulus of continuity on $ Z$, if $ K = {\sup _{f \in Z}}{\max _{x \in I}}\left\vert {f(x)} \right\vert$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0774003-3
PII: S 0002-9939(1985)0774003-3
Keywords: Quasi-smooth functions, modulus of continuity
Article copyright: © Copyright 1985 American Mathematical Society