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An integral representation for the weighted geometric mean and its applications

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Abstract

By virtue of Cauchy’s integral formula in the theory of complex functions, the authors establish an integral representation for the weighted geometric mean, apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function, and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

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Authors and Affiliations

  1. School of Mathematics and Informatics, He’nan Polytechnic University, Jiaozuo, 454010, P. R. China

    Feng Qi

  2. Luoyang, 471000, P. R. China

    Xiao Jing Zhang

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

    Xiao Jing Zhang & Wen Hui Li

Authors
  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, X.J. & Li, W.H. An integral representation for the weighted geometric mean and its applications. Acta. Math. Sin.-English Ser. 30, 61–68 (2014). https://doi.org/10.1007/s10114-013-2547-8

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  • DOI: https://doi.org/10.1007/s10114-013-2547-8

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