Convexity Conditions for Symmetric Borel Derivatives and Borel Smoothness

Abstract

The Borel derivative generalizes classical differentiation by using an integral-based smoothing process. In this paper, monotonicity and convexity conditions for first- and second-order symmetric Borel derivatives are presented. The relation between symmetric Borel derivatives and symmetric Laplace derivatives is also established. Further, Borel smoothness is defined and some basic properties of Borel smooth functions are studied. These results show that several classical shape properties of ordinary derivatives continue to hold in the framework of symmetric Borel derivatives.