Symmetry and the Monotonicity of Certain Riemann Sums

Abstract

We consider conditions ensuring the monotonicity of right and left Riemann sums of a function $f:[0,1]\rightarrow \mathbb {R}$ with respect to uniform partitions. Experimentation suggests that symmetrization may be important and leads us to results such as: if f is decreasing on [0, 1] and its symmetrization, $F(x) := \frac{1}{2}\left( f(x) + f(1-x)\right)$ , is concave then its right Riemann sums increase monotonically with partition size. Applying our results to functions such as $f(x) = 1/\left( 1+x^2\right)$ also leads to a nice application of Descartes’ rule of signs.

This paper is dedicated to the memory and lasting legacy of Jonathan Borwein, son, friend, and colleague.

It was one of the last papers he worked on and exemplifies his experimental approach to mathematical discovery.

Jonathan M. Borwein Passed away suddenly and unexpectedly on 2 August 2016.

Was Laureate Professor and Director of the Centre for Computer-Assisted Research Mathematics and its Applications at the University of Newcastle, Australia.