Optimal Upper and Lower Bounds for Logarithmic Mean*

HOU Shou-w ei ,XU Yan-w u ,CHU Yu-ming

(1.Department of Mathematics,Hangzhou Normal University,Hangzhou 310012,China;2.School of Economics,Shanghai University of Finance and Economic,Shanghai 200433,China;3.Faculty of Science,Huzhou Teachers College,Huzhou 313000,China)

Optimal Upper and Lower Bounds for Logarithmic Mean*

HOU Shou-w ei1,XU Yan-w u2,CHU Yu-ming3

(1.Department of Mathematics,Hangzhou Normal University,Hangzhou 310012,China;2.School of Economics,Shanghai University of Finance and Economic,Shanghai 200433,China;3.Faculty of Science,Huzhou Teachers College,Huzhou 313000,China)

M aking use of elementary differential calculus,we compare the logarithmic mean w ith the convex combination of root-square and harmonic root-squaremeans,and find the greatest valueαand the least valuesβsuch that the double inequalityholds fo r all a,b＞0 w ithare the rootsquare,harmonic root-square,and Logarithmic meansof two positive numbers a and b,w ith a≠b,respectively.

root-squaremean;harmonic root-squaremean;Logarithmic mean

MSC 2000:26E60 26D20

0 In troduction

Fo r p∈R,the p-th pow er mean Mp(a,b)and logarithm ic mean L(a,b)of two positive num bers a and b is defined by respectively.

Recently,both mean values have been the subject of intensive research.In particular,many remarkable inequalities for Mp(a,b)and L(a,b)can be found in the literature[1～17].It iswell know n that Mp(a,b)is continuous and strictly increasing w ith respect to p∈R fo r fixed a,b＞0 w ith a≠b,and many means are special cases of the power mean,for examp le,

are the harmonic root-square,harmonic,geometric,arithmetic,and root-square means of a and b,respectively.

Lin[13]p resent the op timal double inequality

fo r all a,b＞0 w ith a ≠b.

In[6],Long and Chu answer the question:w hat are the greatest value p=p(α,β)and least value q=q(α,β)such that the double inequality

holds fo r all a,b＞0 w ith a≠b andα,β＞0 w ithα+β＜1?

The follow ing sharp bounds for the combination of arithmetic and logarithmic means in term s ofpower mean are given in[7]:

fo r allα∈(0,1)and a,b＞0 w ith a ≠b.

The main purpose of thispaper is to answer the question:w hat are the greatest valueαand least valueβsuch that the double inequality

holds fo r all a,b＞0 w ith a≠b.

1 Lemma

In order to establish our main result we need a lemma,w hich we p resent in this section.

2 Main Result

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MSC 2000:26E60 26D20

對(duì)數(shù)平均的最佳上下界

候守偉1,徐言午2,褚玉明3

(1.杭州師范大學(xué)數(shù)學(xué)系,浙江杭州310012;2.上海財(cái)經(jīng)大學(xué)經(jīng)濟(jì)學(xué)院,上海200433;3.湖州師范學(xué)院理學(xué)院,浙江湖州 313000)

利用初等微分學(xué)比較了對(duì)數(shù)平均與平方根平均和調(diào)和平方根平均的凸組合,發(fā)現(xiàn)了使得雙向不等式αS(a,b)+(1對(duì)所有 a,b＞0且 a≠b成立的α的最大值和β的最小值,其中 S(a,b)分別表示二個(gè)正數(shù) a與b的平方根平均、調(diào)和平方根平均和對(duì)數(shù)平均.

平方根平均;調(diào)和平方根平均;對(duì)數(shù)平均

O174.1

O174.1 Document code:A Article ID:1009-1734(2011)01-0007-04

date:2011-01-21

s:This research is suppo rted by the Natural Science Foundation of China(11071067)and the Innovation Team Foundation of the Department of Education of Zhejiang Porvince(T200924).

Biography:Hou Shou-wei,Postgraduate student of grade 2009,Department of Mathematics,Hangzhou Normal U-niversity,Research Interest:Comp lex Analysis.