Bounding the Difference and Ratio Between the Weighted Arithmetic and Geometric Means

Main Article Content

Feng Qi

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.

Article Details

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.