权方和不等式是一种分式不等式。
a i , b i > 0 {\displaystyle a_{i},b_{i}>0}
{ ∑ i = 1 n a i m + 1 b i m ≥ ( ∑ i = 1 n a i ) m + 1 ( ∑ i = 1 n b i ) m , m < − 1 , m > 0 ∑ i = 1 n a i m + 1 b i m ≤ ( ∑ i = 1 n a i ) m + 1 ( ∑ i = 1 n b i ) m , − 1 < m < 0 {\displaystyle {\begin{cases}\displaystyle \sum _{i=1}^{n}{\frac {a_{i}^{m+1}}{b_{i}^{m}}}\geq {\frac {(\displaystyle \sum _{i=1}^{n}a_{i})^{m+1}}{(\displaystyle \sum _{i=1}^{n}b_{i})^{m}}},m<-1,m>0\\\displaystyle \sum _{i=1}^{n}{\frac {a_{i}^{m+1}}{b_{i}^{m}}}\leq {\frac {(\displaystyle \sum _{i=1}^{n}a_{i})^{m+1}}{(\displaystyle \sum _{i=1}^{n}b_{i})^{m}}},-1<m<0\end{cases}}} [1]
取等条件: a 1 b 1 = a 2 b 2 = . . . = a n b n {\displaystyle {\frac {a_{1}}{b_{1}}}={\frac {a_{2}}{b_{2}}}=...={\frac {a_{n}}{b_{n}}}}
利用伯努利不等式:[1]
s = 1 a 1 + a 2 + . . . + a n , t = 1 b 1 + b 2 + . . . + b n {\displaystyle s={\frac {1}{a_{1}+a_{2}+...+a_{n}}},t={\frac {1}{b_{1}+b_{2}+...+b_{n}}}}
m<-1 或 m>0时
-1<m<0时
∑ i = 1 n a i m + 1 b i m ≤ t m s m + 1 = ( ∑ i = 1 n a i ) m + 1 ( ∑ i = 1 n b i ) m {\displaystyle \sum _{i=1}^{n}{\frac {a_{i}^{m+1}}{b_{i}^{m}}}\leq {\frac {t^{m}}{s^{m+1}}}={\frac {(\displaystyle \sum _{i=1}^{n}a_{i})^{m+1}}{(\displaystyle \sum _{i=1}^{n}b_{i})^{m}}}}