http://en.wikipedia.org/wiki/Jensen's_inequality

If λ1 and λ2 are two arbitrary nonnegative real numbers such that λ1 + λ2 = 1 then convexity of \scriptstyle\varphi implies

\varphi(\lambda_1 x_1+\lambda_2 x_2)\leq \lambda_1\,\varphi(x_1)+\lambda_2\,\varphi(x_2)\text{ for any }x_1,\,x_2.  [这就是凸函数的定义]

This can be easily generalized: if λ1, λ2, ..., λn are nonnegative real numbers such that λ1 + ... + λn = 1, then

\varphi(\lambda_1 x_1+\lambda_2 x_2+\cdots+\lambda_n x_n)\leq \lambda_1\,\varphi(x_1)+\lambda_2\,\varphi(x_2)+\cdots+\lambda_n\,\varphi(x_n),

例如-log(x)是凸函数