Journal of Hebei University (Natural Science Edition) ›› 2017, Vol. 37 ›› Issue (5): 454-456.DOI: 10.3969/j.issn.1000-1565.2017.05.002

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Optimal convex combination bounds for the power mean

MENG Xiangju,TIAN Shuhuan,XU Huifeng   

  1. Department of Mathematics and Computer Science, Baoding College, Baoding 071000, China
  • Received:2016-11-25 Online:2017-09-25 Published:2017-09-25

Abstract: The optimal value of parameters are obtained to make the following double inequality holds for all a,b>0,2Mr(a,b)≥G(a,b)+C(a,b)and A(a,b)+C(a,b)≥2Mr(a,b)where Mr(a,b),G(a,b),C(a,b),A(a,b)denote the power mean,the geometric mean,the contraharmonic mean, the arithmetic mean of two different positive numbers a and b respectively.

Key words: the power mean, the geometric mean, the contraharmonic mean, the arithmetic mean

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