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Some Optimal Convex Combination Bounds for Arithmetic Mean

Gao Hong-ya|Xue Ruihong
경북대학교 자연과학대학 수학과|2014.12|Kyungpook Mathematical Journal|제54권 제4호|pp. 521-529|DOI http://dx.doi.org/10.5666/KMJ.2014.54.4.519

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참고문헌

키워드

arithmetic meanharmonic meangeometric meanOptimal convex combination boundthe second Seiffert mean
주제분류수학

초록

In this paper we derive some optimalconvex combination bounds related to arithmetic mean. We find thegreatest values α1 and α2 and the least values β1 and β2 such that the double inequalities α1 T(a,b)+(1-α1)H(a,b) < A(a,b) < β1T(a,b)+(1-β1)H(a,b) and α2T(a,b)+(1-α2)G(a,b) < A(a,b) < β2T(a,b)+(1-β2)G(a,b) holds for all a,b > 0 with a ≠ 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.

Abstract

In this paper we derive some optimalconvex combination bounds related to arithmetic mean. We find thegreatest values α1 and α2 and the least values β1 and β2 such that the double inequalities α1 T(a,b)+(1-α1)H(a,b) < A(a,b) < β1T(a,b)+(1-β1)H(a,b) and α2T(a,b)+(1-α2)G(a,b) < A(a,b) < β2T(a,b)+(1-β2)G(a,b) holds for all a,b > 0 with a ≠ 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.

참고문헌 (20)

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