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.
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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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Qi, F., Zhang, XJ. & Li, WH. The harmonic and geometric means are Bernstein functions. Bol. Soc. Mat. Mex. 23, 713–736 (2017). https://doi.org/10.1007/s40590-016-0085-y
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DOI: https://doi.org/10.1007/s40590-016-0085-y
Keywords
- Bernstein function
- Harmonic mean
- Geometric mean
- Integral representation
- Stieltjes function
- Induction
- Cauchy integral formula
- Stieltjes-Perron inversion formula