加权算数平均大于等于几何平均
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\sum_{i=1}^{n}p_ix_i \ge \prod_{i=1}^{n} x_{i}^{p_i}(\sum_{i=1}^{n}p_i=1)
∑i=1npixi≥∏i=1nxipi(∑i=1npi=1)
当且仅当
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x1=x2=⋯=xn时取等
证明：
由Jensen不等式
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\ln(\sum_{i=1}^{n}p_ix_i)\ge \sum_{i=1}^{n}p_i\ln x_i=\sum_{i=1}^{n}\ln x_{i}^{p_i}=\ln\prod_{i=1}^{n}x_i^{p_i}
ln(∑i=1npixi)≥∑i=1npilnxi=∑i=1nlnxipi=ln∏i=1nxipi