Majorization理论

验证下面的表达式是服从Schur-convex

\[\frac{{\Delta \left( {\ln {\bf{x}}} \right)}}{{\Delta \left( {\bf{x}} \right)\prod\limits_{i = 1}^N {{x_i}} }} = \frac{{\prod\limits_{i < j} {\left( {\ln {x_i} – \ln {x_j}} \right)} }}{{\prod\limits_{i < j} {\left( {{x_i} – {x_j}} \right)} \prod\limits_{i = 1}^N {{x_i}} }}\]

显然这个函数是对称函数

\[\frac{{\Delta \left( {\ln {\bf{x}}} \right)}}{{\Delta \left( {\bf{x}} \right)\prod\limits_{i = 1}^N {{x_i}} }} = \prod\limits_{i < j} {\frac{{\ln {x_i} – \ln {x_j}}}{{{x_i} – {x_j}}}} \frac{1}{{\prod\limits_{i = 1}^N {{x_i}} }}\]

定义函数

\[f\left( {\bf{x}} \right) = \ln \prod\limits_{i < j} {\frac{{\ln {x_i} – \ln {x_j}}}{{{x_i} – {x_j}}}} \frac{1}{{\prod\limits_{i = 1}^N {{x_i}} }}= \sum\limits_{i = 1}^N {\ln \frac{1}{{{x_i}}} + \sum\limits_{i < j} {\ln \frac{{\ln {x_i} - \ln {x_j}}}{{{x_i} - {x_j}}}} } \]

根据定理[R1. A.3. Theorem.]，等价于证明${f_{\left( 1 \right)}}\left( {\bf{x}} \right) – {f_{\left( 2 \right)}}\left( {\bf{x}} \right)$，${f_{\left( 2 \right)}}\left( {\bf{x}} \right) – {f_{\left( 3 \right)}}\left( {\bf{x}} \right)$，…

发现一个比较好用的不等式

\[\frac{2}{{{x_1} + {x_2}}} < \frac{{\ln {x_1} – \ln {x_2}}}{{{x_1} – {x_2}}} < \frac{1}{2}\left( {\frac{1}{{{x_1}}} + \frac{1}{{{x_2}}}} \right)\]

具体证明过程请参考文献[R2]。

[R1]Marshall, A.W., 1979. Inequalities: Theory of Majorization and Its Applications.

[R2]Outage Analysis of Cascaded RISs Systems Over Doubly-Correlated MIMO Fading Channels. IEEE TWC