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| 对数平均的最佳上下界 | |
| 引用本文： | 候守伟,徐言午,褚玉明.对数平均的最佳上下界[J].湖州师范学院学报,2011,33(1):7-10. |
| 作者姓名： | 候守伟 徐言午 褚玉明 |
| 作者单位： | 1. 杭州师范大学,数学系,浙江,杭州,310012 2. 上海财经大学,经济学院,上海,200433 3. 湖州师范学院,理学院,浙江,湖州,313000 |
| 基金项目： | This researchis supported by the Natural Science Foundation of China(11071067); the Innovation Team Foundation of the Depart ment of Education of Zhejiang Porvince(T200924) |
| 摘 要： | 利用初等微分学比较了对数平均与平方根平均和调和平方根平均的凸组合,发现了使得双向不等式aS(a,b)+(1-a)(H)(a,b)＜L(a,b)＜βS(a,b)+(1-β)(H)(a,b)对所有a,b＞0且a≠b成立的a的最大值和β的最小值,其中S(a,b)=√(a2,b2)/2,(H)(a,b)=√2ab√a2+b2和L(a,b)=(a-b)/(loga-logb)分别表示二个正数a与b的平方根平均、调和平方根平均和对数平均. |
| 关 键 词： | 平方根平均 调和平方根平均 对数平均 |
Optimal Upper and Lower Bounds for Logarithmic Mean |
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| HOU Shou-wei,XU Yan-wu,CHU Yu-ming.Optimal Upper and Lower Bounds for Logarithmic Mean[J].Journal of Huzhou Teachers College,2011,33(1):7-10. | |
| Authors: | HOU Shou-wei XU Yan-wu CHU Yu-ming |
| Institution: | HOU Shou-wei 1,XU Yan-wu 2,CHU Yu-ming 3(1.Department of Mathematics,Hangzhou Normal University,Hangzhou 310012,China,2.School of Economics,Shanghai University of Finance and Economic,Shanghai 200433,3.Faculty of Science,Huzhou Teachers College,Huzhou 313000,China) |
| Abstract: | Making use of elementary differential calculus,we compare the logarithmic mean with the convex combi nation of root-square and harmonic root-square means,and find the greatest value a and the least values β such that the double inequality aS(a,b) + (l-a)(H)(a,B)＜L(a,b)＜βS(a,b) + (1-β)(H)(a,b) holds for all a,b＞0 with a≠b. Here,S(a,b) = √(a2 +b2 )/2 ,(H)(a,b) =√2ab/√a2 +b2 and L(a,b) = (a - b)/(loga-logb) are the root square,harmonic root-square,and Logarithmic means of two positive numbers a and b,with a≠b,respectively. |
| Keywords: | root-square mean harmonic root-square mean Logarithmic mean |
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