Editing Generalized mean (section)

=== Generalized mean inequality ===
{{QM_AM_GM_HM_inequality_visual_proof.svg}}
In general, if {{math|''p'' < ''q''}}, then
<math display=block>M_p(x_1, \dots, x_n) \le M_q(x_1, \dots, x_n)</math>
and the two means are equal if and only if {{math|1= ''x''<sub>1</sub> = ''x''<sub>2</sub> = ... = ''x<sub>n</sub>''}}.

The inequality is true for real values of {{mvar|p}} and {{mvar|q}}, as well as positive and negative infinity values.

It follows from the fact that, for all real {{mvar|p}},
<math display=block>\frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0</math>
which can be proved using [[Jensen's inequality]].

In particular, for {{mvar|p}} in {{math|{−1, 0, 1}<nowiki/>}}, the generalized mean inequality implies the [[Pythagorean means]] inequality as well as the [[inequality of arithmetic and geometric means]].