Some Optimal Convex Combination Bounds for Arithmetic Mean

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 4,  2014, pp.521-529
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.4.521
Title & Authors
Some Optimal Convex Combination Bounds for Arithmetic Mean
Hongya, Gao; Ruihong, Xue;

Abstract
In this paper we derive some optimal convex combination bounds related to arithmetic mean. We find the greatest values $\small{{\alpha}_1}$ and $\small{{\alpha}_2}$ and the least values $\small{{\beta}_1}$ and $\small{{\beta}_2}$ such that the double inequalities $\small{{\alpha}_1T(a,b)+(1-{\alpha}_1)H(a,b)}$$\small{&}$$\small{lt;A(a,b)}$$\small{&}$$\small{lt;{\beta}_1T(a,b)+(1-{\beta}_1)H(a,b)}$ and $\small{{\alpha}_2T(a,b)+(1-{\alpha}_2)G(a,b)}$$\small{&}$$\small{lt;A(a,b)}$$\small{&}$$\small{lt;{\beta}_2T(a,b)+(1-{\beta}_2)G(a,b)}$ holds for all a,b > 0 with $\small{a{\neq}b}$. 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
Optimal convex combination bound;arithmetic mean;harmonic mean;geometric mean;the second Seiffert mean;
Language
English
Cited by
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