凸函数
Jensen Theorem
convex function f \[ f\left( \sum_{i=1}^{n}p_ix_i\right) \le \sum_{i=1}^{n}f(p_ix_i) \] \(x_i \in \mathbf{dom}f\) , \(\mathbf{p} \in\) 概率单纯形
- AM-GM
- \(\sum t_i log x_i \le log \sum t_ix_i\)
others
\(\varphi-mean\)
\(\varphi : (0,+\infty) \rightarrow \mathbf{R}\) continuous and strictly monotonic
\(\varphi-mean\) of a sequence \(a\) (\(a_i\gt0\)): \[ M_\varphi(a)=\varphi^{-1}\left(\sum_{i=1}^{n}p_i\varphi(a_i)\right) \]
when \(M_\psi\) and \(M_\varphi\) comparable?
\(\varphi, \psi:(0,+\infty)\rightarrow \mathbf{R}\) continuous and strictly monotonic s.t. \(\varphi\psi^{-1}\) is \(\left\{\begin{matrix} concave ,& \varphi\;is\;increasing\\ convex ,& \varphi\;is\;decreasing \end{matrix}\right.\) \[ M_\varphi(a)\le M_\psi(a) \] proof \[ M_\varphi(a)=\varphi^{-1}\left(\sum_{i=1}^{n}p_i\varphi(a_i)\right)\\\\ = \varphi^{-1}\left(\sum_{i=1}^{n}p_i\varphi\psi^{-1}(b_i)\right)\\\\ \le \varphi^{-1}\left(\varphi\psi^{-1}\left(\sum_{i=1}^{n}p_ib_i\right)\right) =M_\psi(a) \] for \(\varphi\) is increasing
special case
when \(\varphi(t)=t^r\;(-\infty\lt r \lt +\infty,\; r\ne 0)\) , write \(M_r\) for \(M_\varphi\)
\(M_\infty\) = \(\max_{1\le i\le n} a_i\), \(M_{-\infty}\) = \(\min_{1\le i\le n} a_i\), \(M_0\) = \(\prod{a_i^{p_i}}\),
property
\(M_r\) is continuous monotone increasing function
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