2013/2014 Extreme Results on Certain Generalized Riemann Derivatives
John C. Georgiou
Real Anal. Exchange 40(1): 193-208 (2013/2014).

## 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 $$x \in \mathbb{R}$$?

## Citation

John C. Georgiou. "Extreme Results on Certain Generalized Riemann Derivatives." Real Anal. Exchange 40 (1) 193 - 208, 2013/2014.

## Information

Published: 2013/2014
First available in Project Euclid: 1 July 2015

zbMATH: 06848831
MathSciNet: MR3365398

Subjects:
Primary: 26A24 , 26A27 , 26A51‎ , 26B08
Secondary: 11A55 , 40A30 , 54C50

Keywords: convexity , divided differences , Generalized derivatives , non-differentiability