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The harmonic and geometric means are Bernstein functions

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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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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan Province, China

    Feng Qi

  2. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia Autonomous Region, 028043, China

    Feng Qi

  3. Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Feng Qi & Wen-Hui Li

  4. The 59th Middle School, Jianxi District, Luoyang, Henan Province, 471000, China

    Xiao-Jing Zhang

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  1. Feng Qi
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  2. Xiao-Jing Zhang
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  3. Wen-Hui Li
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Correspondence to Feng Qi.

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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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