Some conditions for $n$-convex functions

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) \[ {F_{(r)}}(x) = \lim \limits _{h \to 0} \left ( {\frac {{r!}} {{{h^r}}}} \right )\left ( {F(x + h) - F(x) - \sum \limits _{k = 1}^{r - 1} {\frac {{{h^k}{F_{(k)}}(x)}} {{k!}}} } \right )\] 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., \[ \lim \limits _{h \to 0} \left ( {\frac {{n!}} {{{h^{n - 1}}}}} \right )\left [ {\frac {{F(x + h) + F(x - h)}} {2} - \sum \limits _{k = 0}^{n/2 - 1} {\frac {{{h^{2k}}{D^{2k}}F(x)}} {{(2k)!}}} } \right ] = 0\] 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.