权方和不等式是一个数学中重要的不等式。
其证明需要用到赫尔德不等式(Holder),可用于放缩的方法求最值(极值)、证明不等式等。权方和不等式的证明如下：设$a_i,b_i\\in R_+$($i=1；2,...,n$),则$\\sum_{i=1}^{n}{\\frac{a_{i}^{m+1}}{b_{i}^{m}}}=a^{m+1}\\sum_{i=1}^{n}{\\frac{1}{b_{im}}}-a^{m}sum_{i=1}^{n}{frac{1}{a_{im}}}$。令$A=\\sum_{i=1}^{n}{\\frac{a_{i}^{m+1}}{b_{i}^{m}}}$,$B=\\sum_{i=1}^{n}{\\frac{1}{b_{im}}}$,$C=\\sum_{i=1}^{n}{\\frac{1}{a_{im}}}$,则有$A=B-C$,且$B>0$,$C>0$。由赫尔德不等式可知：$\\sum_{i=1}^{n}{\\frac{1}{b_{im}}}\\geq \\sum_{i=1}^{n}{\\frac{1}{a_{im}}}$,$sum_{i=1}^{n}{frac{1}{a_{im}}}geq \\sum_{i=1}^{n}{\\frac{1}{b_{im}}}$。因此，我们有：$$A=B-C\\leq B-\\sum_{i=1}^{n}{\\frac{1}{a_{im}}}\\leq B+\\sum_{i=1}^{n}{\\frac{1}{b_{im}}}-B=\\sum_{i=1}^{n}{\\frac{1}{b_{im}}}-C\\leq \\sum_{i=1}^{n}{\\frac{1}{b_{im}}}=sum_{i=1}^{n}{frac{a^{m+1}}{b^{m}}}=A$$
