Symmetry and the Monotonicity of Certain Riemann Sums

  • David Borwein
  • Jonathan M. Borwein
  • Brailey SimsEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 313)


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.


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • David Borwein
    • 1
  • Jonathan M. Borwein
    • 2
  • Brailey Sims
    • 2
    Email author
  1. 1.Department of MathematicsWestern UniversityLondonCanada
  2. 2.School of Mathematical and Physical SciencesThe Centre for Computer Assisted Research Mathematics and its Applications (CARMA), The University of NewcastleCallaghanAustralia

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