Editing Generalized mean (section)

===Equivalence of inequalities between means of opposite signs===
Suppose an average between power means with exponents {{mvar|p}} and {{mvar|q}} holds:
<math display="block">\left(\sum_{i=1}^n w_i x_i^p\right)^{1/p} \geq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}</math>
applying this, then:
<math display="block">\left(\sum_{i=1}^n\frac{w_i}{x_i^p}\right)^{1/p} \geq \left(\sum_{i=1}^n\frac{w_i}{x_i^q}\right)^{1/q}</math>

We raise both sides to the power of −1 (strictly decreasing function in positive reals):
<math display="block">\left(\sum_{i=1}^nw_ix_i^{-p}\right)^{-1/p}
= \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}\right)^{1/p}
\leq \left(\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}\right)^{1/q}
= \left(\sum_{i=1}^nw_ix_i^{-q}\right)^{-1/q}</math>

We get the inequality for means with exponents {{math|−''p''}} and {{math|−''q''}}, and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.