Editing Generalized mean (section)

===Geometric mean===
For any {{math|''q'' > 0}} and non-negative weights summing to 1, the following inequality holds:
<math display="block">\left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q} \leq \prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.</math>

The proof follows from [[Jensen's inequality]], making use of the fact the [[logarithm]] is concave:
<math display=block>\log \prod_{i=1}^n x_i^{w_i} = \sum_{i=1}^n w_i\log x_i \leq \log \sum_{i=1}^n w_i x_i.</math>

By applying the [[exponential function]] to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get
<math display=block>\prod_{i=1}^n x_i^{w_i} \leq \sum_{i=1}^n w_i x_i.</math>

Taking {{mvar|q}}-th powers of the {{mvar|x<sub>i</sub>}} yields
<math display=block>\begin{align}
&\prod_{i=1}^n x_i^{q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^q \\
&\prod_{i=1}^n x_i^{w_i} \leq \left(\sum_{i=1}^n w_i x_i^q\right)^{1/q}.\end{align}</math>

Thus, we are done for the inequality with positive {{mvar|q}}; the case for negatives is identical but for the swapped signs in the last step:

<math display=block>\prod_{i=1}^n x_i^{-q{\cdot}w_i} \leq \sum_{i=1}^n w_i x_i^{-q}.</math>

Of course, taking each side to the power of a negative number {{math|-1/''q''}} swaps the direction of the inequality.

<math display=block>\prod_{i=1}^n x_i^{w_i} \geq \left(\sum_{i=1}^n w_i x_i^{-q}\right)^{-1/q}.</math>