| 摘要: |
| 众所周知，两个正实数a与b的p次幂平均函数为Mp(a,b)=〖JB({〗(〖SX(〗(ap+bp)〖〗2〖SX)〗)〖SX(〗1〖〗p〖SX)〗,p≠0〖KF(〗ab〖KF)〗,p=0〖JB)〗 .证明了当p≤0时，幂平均函数Mp(a,b)关于参变数p是凸的; 进一步，在p≤0和p≥0时，幂平均函数Mp(a,b)关于参变数p还分别是对数凸及对数凹的. |
| 关键词: 幂平均函数 对数凸性 偏导数 |
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| Discussion on Logarithm Convexity of pth Power Mean Function |
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GUAN Chun mei
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| Abstract: |
| It is well known that the pth power mean function of two positive real number a and b is defined as Mp(a,b)=((ap+bp)/2)1/p,p≠0 while Mp(a,b)=〖KF(〗ab,〖KF)〗p=0. This paper proves that power mean function Mp(a,b) is convex for parameter p≤0, that furthermore, power mean function Mp(a,b) is log convex for parameter p≤0 and is log concave for parameter p≥0 respectively. |
| Key words: power mean function logarithmic convexity partial derivative |
