Title: Bounding the Difference and Ratio Between the Weighted Arithmetic and Geometric Means
Author(s): Feng Qi
Pages: 132-135
Cite as:
Feng Qi, Bounding the Difference and Ratio Between the Weighted Arithmetic and Geometric Means, Int. J. Anal. Appl., 13 (2) (2017), 132-135.

Abstract


In the paper, making use of two integral representations for the difference and ratio of the weighted arithmetic and geometric means and employing the weighted arithmetic-geometric-harmonic mean inequality, the author bounds the difference and ratio between the weighted arithmetic and geometric means in the form of double inequalities.

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References


  1. B.-N. Guo and F. Qi, On the degree of the weighted geometric mean as a complete Bernstein function, Afr. Mat. 26 (7) (2015), 1253–1262.

  2. F. Qi and B.-N. Guo, The reciprocal of the geometric mean of many positive numbers is a Stieltjes transform, J. Comput. Appl. Math. 311 (2017), 165–170.

  3. F. Qi and B.-N. Guo, The reciprocal of the weighted geometric mean is a Stieltjes function, Bol. Soc. Mat. Mex. (2016). doi:10.1007/s40590-016-0151-5.

  4. F. Qi and B.-N. Guo, The reciprocal of the weighted geometric mean of many positive numbers is a Stieltjes function, ResearchGate Working Paper (2016), doi:10.13140/RG.2.2.23822.36163.

  5. F. Qi, B.-N. Guo, V. Cernanova, and X.-T. Shi, Explicit expressions, Cauchy products, integral representations, convexity, and inequalities of central Delannoy numbers, ResearchGate Working Paper (2016), doi:10.13140/RG.2.1.4889.6886.

  6. F. Qi, X.-J. Zhang, and W.-H. Li, An elementary proof of the weighted geometric mean being a Bernstein function, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 77 (1) (2015), 35–38.

  7. F. Qi, X.-J. Zhang, and W.-H. Li, An integral representation for the weighted geometric mean and its applications, Acta Math. Sin. (Engl. Ser.) 30 (1) (2014), 61–68.

  8. F. Qi, X.-J. Zhang, and W.-H. Li, L´ evy-Khintchine representation of the geometric mean of many positive numbers and applications, Math. Inequal. Appl. 17 (2) (2014), 719–729.

  9. F. Qi, X.-J. Zhang, and W.-H. Li, L´ evy-Khintchine representations of the weighted geometric mean and the logarithmic mean, Mediterr. J. Math. 11 (2) (2014), 315–327.

  10. F. Qi, X.-J. Zhang, and W.-H. Li, The harmonic and geometric means are Bernstein functions, Bol. Soc. Mat. Mex. (2016), doi:10.1007/s40590-016-0085-y.