<s>
In	O
mathematics	O
,	O
generalized	B-Language
means	I-Language
(	O
or	O
power	B-Language
mean	I-Language
or	O
Hölder	B-Language
mean	I-Language
from	O
Otto	O
Hölder	O
)	O
are	O
a	O
family	O
of	O
functions	O
for	O
aggregating	O
sets	O
of	O
numbers	O
.	O
</s>
<s>
If	O
is	O
a	O
non-zero	O
real	O
number	O
,	O
and	O
are	O
positive	O
real	O
numbers	O
,	O
then	O
the	O
generalized	B-Language
mean	I-Language
or	O
power	B-Language
mean	I-Language
with	O
exponent	O
of	O
these	O
positive	O
real	O
numbers	O
is	O
:	O
</s>
<s>
Furthermore	O
,	O
for	O
a	O
sequence	O
of	O
positive	O
weights	O
we	O
define	O
the	O
weighted	O
power	B-Language
mean	I-Language
as	O
:	O
</s>
<s>
In	O
particular	O
,	O
for	O
in	O
,	O
the	O
generalized	B-Language
mean	I-Language
inequality	I-Language
implies	O
the	O
Pythagorean	O
means	O
inequality	O
as	O
well	O
as	O
the	O
inequality	O
of	O
arithmetic	O
and	O
geometric	O
means	O
.	O
</s>
<s>
We	O
will	O
prove	O
the	O
weighted	O
power	B-Language
mean	I-Language
inequality	O
.	O
</s>
<s>
The	O
proof	O
for	O
unweighted	O
power	B-Language
means	I-Language
can	O
be	O
easily	O
obtained	O
by	O
substituting	O
.	O
</s>
<s>
Suppose	O
an	O
average	O
between	O
power	B-Language
means	I-Language
with	O
exponents	O
and	O
holds	O
:	O
</s>
<s>
The	O
power	B-Language
mean	I-Language
could	O
be	O
generalized	O
further	O
to	O
the	O
generalized	O
-mean	O
:	O
</s>
<s>
The	O
power	B-Language
mean	I-Language
is	O
obtained	O
for	O
.	O
</s>
<s>
A	O
power	B-Language
mean	I-Language
serves	O
a	O
non-linear	O
moving	O
average	O
which	O
is	O
shifted	O
towards	O
small	O
signal	O
values	O
for	O
small	O
and	O
emphasizes	O
big	O
signal	O
values	O
for	O
big	O
.	O
</s>
<s>
Given	O
an	O
efficient	O
implementation	O
of	O
a	O
moving	B-Algorithm
arithmetic	I-Algorithm
mean	I-Algorithm
called	O
smooth	O
one	O
can	O
implement	O
a	O
moving	O
power	B-Language
mean	I-Language
according	O
to	O
the	O
following	O
Haskell	B-Language
code	O
.	O
</s>
<s>
For	O
big	O
it	O
can	O
serve	O
as	O
an	O
envelope	B-Algorithm
detector	I-Algorithm
on	O
a	O
rectified	B-Algorithm
signal	O
.	O
</s>
