The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

For p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(log⁡b-log⁡a), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In t...

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Main Authors:	Wei-Feng Xia, Yu-Ming Chu, Gen-Di Wang
Format:	Article
Language:	English
Published:	Hindawi Limited 2010-01-01
Series:	Abstract and Applied Analysis
Online Access:	http://dx.doi.org/10.1155/2010/604804

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spelling	doaj-3812b5f08ee54703a2523d23dd7ecde72020-11-24T22:37:40ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092010-01-01201010.1155/2010/604804604804The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic MeansWei-Feng Xia0Yu-Ming Chu1Gen-Di Wang2School of Teacher Education, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaDepartment of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaDepartment of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, ChinaFor p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(log⁡b-log⁡a), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In this paper, we answer the question: for α∈(0,1), what are the greatest value p and the least value q, such that the double inequality Mp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b) holds for all a,b>0?http://dx.doi.org/10.1155/2010/604804
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Wei-Feng Xia Yu-Ming Chu Gen-Di Wang
spellingShingle	Wei-Feng Xia Yu-Ming Chu Gen-Di Wang The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means Abstract and Applied Analysis
author_facet	Wei-Feng Xia Yu-Ming Chu Gen-Di Wang
author_sort	Wei-Feng Xia
title	The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
title_short	The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
title_full	The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
title_fullStr	The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
title_full_unstemmed	The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means
title_sort	optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means
publisher	Hindawi Limited
series	Abstract and Applied Analysis
issn	1085-3375 1687-0409
publishDate	2010-01-01
description	For p∈ℝ, the power mean Mp(a,b) of order p, logarithmic mean L(a,b), and arithmetic mean A(a,b) of two positive real values a and b are defined by Mp(a,b)=((ap+bp)/2)1/p, for p≠0 and Mp(a,b)=ab, for p=0, L(a,b)=(b-a)/(log⁡b-log⁡a), for a≠b and L(a,b)=a, for a=b and A(a,b)=(a+b)/2, respectively. In this paper, we answer the question: for α∈(0,1), what are the greatest value p and the least value q, such that the double inequality Mp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b) holds for all a,b>0?
url	http://dx.doi.org/10.1155/2010/604804
work_keys_str_mv	AT weifengxia theoptimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans AT yumingchu theoptimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans AT gendiwang theoptimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans AT weifengxia optimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans AT yumingchu optimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans AT gendiwang optimalupperandlowerpowermeanboundsforaconvexcombinationofthearithmeticandlogarithmicmeans
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The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

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