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''for series''
 
''for series''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463404.png" /> then
+
If $  p > 1 $,  
 +
$  a _ {n} \geq  0 $
 +
and $  A _ {n} = a _ {1} + \dots + a _ {n} $,
 +
$  n = 1, 2 \dots $
 +
then
 +
 
 +
$$
 +
\sum _ {n = 1 } ^  \infty 
 +
\left (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463405.png" /></td> </tr></table>
+
\frac{A _ {n} }{n}
  
except when all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463406.png" /> are zero. The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463407.png" /> in this inequality is best possible.
+
\right )  ^ {p}  < \
 +
\left (
 +
\frac{p}{p - 1 }
 +
 
 +
\right )  ^ {p}
 +
\sum _ {n = 1 } ^  \infty 
 +
a _ {n}  ^ {p} ,
 +
$$
 +
 
 +
except when all the $  a _ {n} $
 +
are zero. The constant $  ( p/( p - 1))  ^ {p} $
 +
in this inequality is best possible.
  
 
The Hardy inequalities for integrals are:
 
The Hardy inequalities for integrals are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463408.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
x  ^ {-} p \left |
 +
\int\limits _ { 0 } ^ { x }  f ( t)  dt \right |  ^ {p}
 +
dx  < \left (
 +
 
 +
\frac{p}{p - 1 }
 +
 
 +
\right )  ^ {p}
 +
\int\limits _ { 0 } ^  \infty 
 +
| f ( x) |  ^ {p}  dx,\ \
 +
p > 1 ,
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h0463409.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
\left |
 +
\int\limits _ { x } ^  \infty 
 +
f ( t)  dt \right |  ^ {p}
 +
dx  < p  ^ {p}
 +
\int\limits _ { 0 } ^  \infty 
 +
| xf ( x) |  ^ {p}  dx,\ \
 +
p > 1.
 +
$$
  
The inequalities are valid for all functions for which the right-hand sides are finite, except when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634010.png" /> vanishes almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634011.png" />. (In this case the inequalities turn into equalities.) The constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634013.png" /> are best possible.
+
The inequalities are valid for all functions for which the right-hand sides are finite, except when $  f $
 +
vanishes almost-everywhere on $  ( 0, + \infty ) $.  
 +
(In this case the inequalities turn into equalities.) The constants $  ( p/( p - 1))  ^ {p} $
 +
and $  p  ^ {p} $
 +
are best possible.
  
 
The integral Hardy inequalities can be generalized to arbitrary intervals:
 
The integral Hardy inequalities can be generalized to arbitrary intervals:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634014.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }
 +
\left | x ^ {\alpha - 1 }
 +
\int\limits _ { a } ^ { x }  f ( t)  dt \right |  ^ {p}
 +
dx  \leq  c
 +
\int\limits _ { a } ^ { b }
 +
| x  ^  \alpha  f ( x) |  ^ {p}  dx,\ \
 +
\alpha < 1 - {
 +
\frac{1}{p}
 +
} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634015.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }  \left | x ^ {\alpha - 1 } \int\limits _ { x } ^ { b }  f ( t)  dt \right |  ^ {p}  dx  \leq  c
 +
\int\limits _ { a } ^ { b }  | x  ^  \alpha  f ( x) |  ^ {p}  dx,\  \alpha > 1 - {
 +
\frac{1}{p}
 +
} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634017.png" />, and where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634018.png" />'s are certain constants.
+
where $  0 \leq  a < b \leq  + \infty $,  
 +
$  1 < p < + \infty $,  
 +
and where the $  c $'
 +
s are certain constants.
  
 
Generalized Hardy inequalities are inequalities of the form
 
Generalized Hardy inequalities are inequalities of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits _ { a } ^ { b }
 +
\left | \phi ( x)
 +
\int\limits _ { a } ^ { x }
 +
f ( t)  dt \right |  ^ {p} \
 +
dx  \leq  c
 +
\int\limits _ { a } ^ { b }
 +
| \psi ( x) f ( x) |  ^ {p}  dx,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { a } ^ { b }  \left | \phi ( x) \int\limits _ { x } ^ { b }  f ( t)
 +
dt \right |  ^ {p}  dx  \leq  c \int\limits _ { a } ^ { b }  | \psi ( x) f ( x) |  ^ {p}  dx.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634022.png" />, inequality (1) holds if and only if
+
If $  a = 0 $
 +
and $  b = + \infty $,  
 +
inequality (1) holds if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634023.png" /></td> </tr></table>
+
$$
 +
\sup _ {x > 0 }
 +
\left [ \int\limits _ { x } ^  \infty 
 +
| \phi ( t) |  ^ {p}  dt
 +
\right ]  ^ {1/p} \left [
 +
\int\limits _ { 0 } ^ { x }
 +
| \psi ( t) | ^ {- p  ^  \prime  }
 +
dt \right ] ^ {1/p  ^  \prime  }
 +
< + \infty ,
 +
$$
  
 
and (2) holds if and only if
 
and (2) holds if and only if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634024.png" /></td> </tr></table>
+
$$
 +
\sup _ {x > 0 }
 +
\left [ \int\limits _ { 0 } ^ { x }
 +
| \phi ( t) |  ^ {p} \
 +
dt \right ]  ^ {1/p}
 +
\left [ \int\limits _ { x } ^  \infty 
 +
| \psi ( t) | ^ {- p  ^  \prime  } \
 +
dt \right ] ^ {1/p  ^  \prime  }
 +
< + \infty ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046340/h04634025.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{p}
 +
} + {
 +
\frac{1}{p  ^  \prime  }
 +
= 1.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Muckenhoupt,  "Hardy's inequality with weights"  ''Studia Math.'' , '''44'''  (1972)  pp. 31–38</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Muckenhoupt,  "Hardy's inequality with weights"  ''Studia Math.'' , '''44'''  (1972)  pp. 31–38</TD></TR></table>

Latest revision as of 19:43, 5 June 2020


for series

If $ p > 1 $, $ a _ {n} \geq 0 $ and $ A _ {n} = a _ {1} + \dots + a _ {n} $, $ n = 1, 2 \dots $ then

$$ \sum _ {n = 1 } ^ \infty \left ( \frac{A _ {n} }{n} \right ) ^ {p} < \ \left ( \frac{p}{p - 1 } \right ) ^ {p} \sum _ {n = 1 } ^ \infty a _ {n} ^ {p} , $$

except when all the $ a _ {n} $ are zero. The constant $ ( p/( p - 1)) ^ {p} $ in this inequality is best possible.

The Hardy inequalities for integrals are:

$$ \int\limits _ { 0 } ^ \infty x ^ {-} p \left | \int\limits _ { 0 } ^ { x } f ( t) dt \right | ^ {p} dx < \left ( \frac{p}{p - 1 } \right ) ^ {p} \int\limits _ { 0 } ^ \infty | f ( x) | ^ {p} dx,\ \ p > 1 , $$

and

$$ \int\limits _ { 0 } ^ \infty \left | \int\limits _ { x } ^ \infty f ( t) dt \right | ^ {p} dx < p ^ {p} \int\limits _ { 0 } ^ \infty | xf ( x) | ^ {p} dx,\ \ p > 1. $$

The inequalities are valid for all functions for which the right-hand sides are finite, except when $ f $ vanishes almost-everywhere on $ ( 0, + \infty ) $. (In this case the inequalities turn into equalities.) The constants $ ( p/( p - 1)) ^ {p} $ and $ p ^ {p} $ are best possible.

The integral Hardy inequalities can be generalized to arbitrary intervals:

$$ \int\limits _ { a } ^ { b } \left | x ^ {\alpha - 1 } \int\limits _ { a } ^ { x } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | x ^ \alpha f ( x) | ^ {p} dx,\ \ \alpha < 1 - { \frac{1}{p} } , $$

$$ \int\limits _ { a } ^ { b } \left | x ^ {\alpha - 1 } \int\limits _ { x } ^ { b } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | x ^ \alpha f ( x) | ^ {p} dx,\ \alpha > 1 - { \frac{1}{p} } , $$

where $ 0 \leq a < b \leq + \infty $, $ 1 < p < + \infty $, and where the $ c $' s are certain constants.

Generalized Hardy inequalities are inequalities of the form

$$ \tag{1 } \int\limits _ { a } ^ { b } \left | \phi ( x) \int\limits _ { a } ^ { x } f ( t) dt \right | ^ {p} \ dx \leq c \int\limits _ { a } ^ { b } | \psi ( x) f ( x) | ^ {p} dx, $$

$$ \tag{2 } \int\limits _ { a } ^ { b } \left | \phi ( x) \int\limits _ { x } ^ { b } f ( t) dt \right | ^ {p} dx \leq c \int\limits _ { a } ^ { b } | \psi ( x) f ( x) | ^ {p} dx. $$

If $ a = 0 $ and $ b = + \infty $, inequality (1) holds if and only if

$$ \sup _ {x > 0 } \left [ \int\limits _ { x } ^ \infty | \phi ( t) | ^ {p} dt \right ] ^ {1/p} \left [ \int\limits _ { 0 } ^ { x } | \psi ( t) | ^ {- p ^ \prime } dt \right ] ^ {1/p ^ \prime } < + \infty , $$

and (2) holds if and only if

$$ \sup _ {x > 0 } \left [ \int\limits _ { 0 } ^ { x } | \phi ( t) | ^ {p} \ dt \right ] ^ {1/p} \left [ \int\limits _ { x } ^ \infty | \psi ( t) | ^ {- p ^ \prime } \ dt \right ] ^ {1/p ^ \prime } < + \infty , $$

$$ { \frac{1}{p} } + { \frac{1}{p ^ \prime } } = 1. $$

References

[1] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[2] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[3] B. Muckenhoupt, "Hardy's inequality with weights" Studia Math. , 44 (1972) pp. 31–38
How to Cite This Entry:
Hardy inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hardy_inequality&oldid=47176
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article