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

 

Some conditions for $ n$-convex functions


Author: G. E. Cross
Journal: Proc. Amer. Math. Soc. 82 (1981), 587-592
MSC: Primary 26A51
MathSciNet review: 614883
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Abstract: It is shown that if $ F$ is a function defined and continuous on $ [a,b]$ such that (where $ n$ is an even integer):

(a)

$\displaystyle {F_{(r)}}(x) = \mathop {\lim }\limits_{h \to 0} \left( {\frac{{r!... ...(x) - \sum\limits_{k = 1}^{r - 1} {\frac{{{h^k}{F_{(k)}}(x)}} {{k!}}} } \right)$

exists and is finite in $ (a,b)$ for $ 1 \leqslant r \leqslant n - 2$;

(b) for $ x \in (a,b)\backslash E$, where $ E$ is countable, $ F$ is $ n$-smooth, i.e.,

$\displaystyle \mathop {\lim }\limits_{h \to 0} \left( {\frac{{n!}} {{{h^{n - 1}... ...imits_{k = 0}^{n/2 - 1} {\frac{{{h^{2k}}{D^{2k}}F(x)}} {{(2k)!}}} } \right] = 0$

where $ {D^{2k}}F(x)$ denotes the (symmetric) de la Vallée-Poussin derivative;

(c) $ {\bar D^n}F(x) \geqslant 0$ a.e. in $ (a,b)$;

(d) $ {\bar D^n}F(x) > - \infty $ for $ x \in (a,b)\backslash S$ where $ S$ is countable and $ F(x)$ is $ n$-smooth in $ S$; then $ F(x)$ is $ n$-convex in $ [a,b]$. The same result holds for $ n$ odd. This is an improvement on the known result when $ S$ is a scattered set.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0614883-1
PII: S 0002-9939(1981)0614883-1
Keywords: Convex function
Article copyright: © Copyright 1981 American Mathematical Society