## Real Analysis Exchange

### Extreme Results on Certain Generalized Riemann Derivatives

John C. Georgiou

#### Abstract

In this paper the following question is investigated. Given a natural number $r$ and numbers $\alpha_j,\beta_j$ for $j=0,1,\dots,r$ satisfying $\alpha_0 <\alpha_1 < \dots \lt \alpha_r$ and \begin{equation*} \sum_{j=0}^{r} \beta_j \alpha_j^k= \begin{cases} 0 & \text{if $k=0,1,\dots,r-1$}\\ r!& \text{if $k=r$ } \end{cases} , \end{equation*} is there a $2\pi$-periodic, $r-1$ times continuously differentiable function $f$ such that \begin{equation*} \limsup_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \limsup_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \infty, \end{equation*} \begin{equation*} \liminf_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \liminf_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = - \infty \end{equation*} for every $x \in \mathbb{R}$?

#### Article information

Source
Real Anal. Exchange, Volume 40, Number 1 (2015), 193-208.

Dates
First available in Project Euclid: 1 July 2015