Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Some Optimal Convex Combination Bounds for Arithmetic Mean

  • Hongya, Gao (College of Mathematics and Computer Science, Hebei University) ;
  • Ruihong, Xue (College of Mathematics and Computer Science, Hebei University)
  • Received : 2011.06.09
  • Accepted : 2013.04.29
  • Published : 2014.12.23
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Abstract

In this paper we derive some optimal convex combination bounds related to arithmetic mean. We find the greatest values ${\alpha}_1$ and ${\alpha}_2$ and the least values ${\beta}_1$ and ${\beta}_2$ such that the double inequalities $${\alpha}_1T(a,b)+(1-{\alpha}_1)H(a,b)<A(a,b)<{\beta}_1T(a,b)+(1-{\beta}_1)H(a,b)$$ and $${\alpha}_2T(a,b)+(1-{\alpha}_2)G(a,b)<A(a,b)<{\beta}_2T(a,b)+(1-{\beta}_2)G(a,b)$$ holds for all a,b > 0 with $a{\neq}b$. Here T(a,b), H(a,b), A(a,b) and G(a,b) denote the second Seiffert, harmonic, arithmetic and geometric means of two positive numbers a and b, respectively.

Keywords

Acknowledgement

Supported by : NSF of Hebei Province

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