Editing Generalized mean (section)

==Properties==

Let <math>x_1, \dots, x_n</math> be a sequence of positive real numbers, then the following properties hold:<ref name=sykora>{{cite journal|last=Sýkora|first=Stanislav|year=2009|title=Mathematical means and averages: basic properties|journal=Stan's Library |location=Castano Primo, Italy|volume=III |doi=10.3247/SL3Math09.001 }}</ref>

#<math>\min(x_1, \dots, x_n) \le M_p(x_1, \dots, x_n) \le \max(x_1, \dots, x_n)</math>.<!--
-->{{block indent|left=1|text= Each generalized mean always lies between the smallest and largest of the {{mvar|x}} values.}}
#<math>M_p(x_1, \dots, x_n) = M_p(P(x_1, \dots, x_n))</math>, where <math>P</math> is a permutation operator.<!--
-->{{block indent|left=1|text= Each generalized mean is a symmetric function of its arguments; permuting the arguments of a generalized mean does not change its value.}}
#<math>M_p(b x_1, \dots, b x_n) = b \cdot M_p(x_1, \dots, x_n)</math>.<!--
-->{{block indent|left=1|text= Like most [[Mean#Properties|mean]]s, the generalized mean is a [[homogeneous function]] of its arguments {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}. That is, if {{mvar|b}} is a positive real number, then the generalized mean with exponent {{mvar|p}} of the numbers <math>b\cdot x_1,\dots, b\cdot x_n</math> is equal to {{mvar|b}} times the generalized mean of the numbers {{math|''x''<sub>1</sub>, ..., ''x<sub>n</sub>''}}.}}
#<math>M_p(x_1, \dots, x_{n \cdot k}) = M_p\left[M_p(x_1, \dots, x_{k}), M_p(x_{k + 1}, \dots, x_{2 \cdot k}), \dots, M_p(x_{(n - 1) \cdot k + 1}, \dots, x_{n \cdot k})\right]</math>.<!--
-->{{block indent|left=1|text= Like the [[quasi-arithmetic mean]]s, the computation of the mean can be split into computations of equal sized sub-blocks. This enables use of a [[divide and conquer algorithm]] to calculate the means, when desirable.}}

=== 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]].