Boletín de la Sociedad Matemática Mexicana

, Volume 23, Issue 2, pp 713–736 | Cite as

The harmonic and geometric means are Bernstein functions

Original Article

Abstract

In the paper, the authors present by several approaches that both the harmonic mean and the geometric mean of two positive numbers are Bernstein functions and establish their integral representations.

Keywords

Bernstein function Harmonic mean Geometric mean  Integral representation Stieltjes function Induction  Cauchy integral formula Stieltjes-Perron inversion formula 

Mathematics Subject Classification

Primary 26E60 Secondary 26A48 30E20 44A10 65R10 

Notes

Acknowledgments

This first author was partially supported by the National Natural Science Foundation of China under Grant No. 11361038 and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY14192, China. The authors thank anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

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Copyright information

© Sociedad Matemática Mexicana 2016

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliaoChina
  3. 3.Department of Mathematics, School of ScienceTianjin Polytechnic UniversityTianjinChina
  4. 4.The 59th Middle SchoolLuoyangChina

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