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''in the simplest discrete form''
 
''in the simplest discrete form''
  
 
The inequality
 
The inequality
  
<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/j/j054/j054220/j0542201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f ( \lambda _ {1} x _ {1} + \dots +
 +
\lambda _ {n} x _ {n} )  \leq  \
 +
\lambda _ {1} f ( x _ {1} ) + \dots +
 +
\lambda _ {n} f ( x _ {n} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542202.png" /> is a convex function on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542204.png" /> (see [[Convex function (of a real variable)|Convex function (of a real variable)]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542207.png" />, and
+
where $  f $
 +
is a convex function on some set $  C $
 +
in $  \mathbf R $(
 +
see [[Convex function (of a real variable)|Convex function (of a real variable)]]), $  x _ {i} \in C $,  
 +
$  \lambda _ {i} \geq  0 $,  
 +
$  i = 1 \dots n $,  
 +
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/j/j054/j054220/j0542208.png" /></td> </tr></table>
+
$$
 +
\lambda _ {1} + \dots + \lambda _ {n}  = 1.
 +
$$
  
Equality holds if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j0542209.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422010.png" /> is linear. Jensen's integral inequality for a convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422011.png" /> is:
+
Equality holds if and only if $  x _ {1} = \dots = x _ {n} $
 +
or if $  f $
 +
is linear. Jensen's integral inequality for a convex function $  f $
 +
is:
  
<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/j/j054/j054220/j05422012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f \left ( \int\limits _ { D } \lambda ( t) x ( t)  dt
 +
\right )  \leq  \int\limits _ { D }
 +
\lambda ( t) f ( x ( t)) dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422015.png" /> and
+
where $  x ( D) \subset  C $,  
 +
$  \lambda ( t) \geq  0 $
 +
for $  t \in D $
 +
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/j/j054/j054220/j05422016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { D } \lambda ( t)  dt  = 1.
 +
$$
  
Equality holds if and only if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422018.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422019.png" /> is linear on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422021.png" /> is a concave function, the inequality signs in (1) and (2) must be reversed. Inequality (1) was established by O. Hölder, and (2) by J.L. Jensen [[#References|[2]]].
+
Equality holds if and only if either $  x ( t) = \textrm{ const } $
 +
on $  D $
 +
or if $  f $
 +
is linear on $  x ( D) $.  
 +
If $  f $
 +
is a concave function, the inequality signs in (1) and (2) must be reversed. Inequality (1) was established by O. Hölder, and (2) by J.L. Jensen [[#References|[2]]].
  
With suitable choices of the convex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422022.png" /> and the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422023.png" /> or weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422024.png" />, inequalities (1) and (2) become concrete inequalities, among which one finds the majority of the classical inequalities. For example, if in (1) one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422026.png" />, then one obtains an inequality between the weighted [[Arithmetic mean|arithmetic mean]] and the [[Geometric mean|geometric mean]]:
+
With suitable choices of the convex function $  f $
 +
and the weights $  \lambda _ {i} $
 +
or weight function $  \lambda $,  
 +
inequalities (1) and (2) become concrete inequalities, among which one finds the majority of the classical inequalities. For example, if in (1) one sets $  f( x) = -  \mathop{\rm ln}  x $,
 +
$  x > 0 $,  
 +
then one obtains an inequality between the weighted [[Arithmetic mean|arithmetic mean]] and the [[Geometric mean|geometric mean]]:
  
<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/j/j054/j054220/j05422027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
x _ {1} ^ {\lambda _ {1} } \dots
 +
x _ {n} ^ {\lambda _ {n} }  \leq  \
 +
\lambda _ {1} x _ {1} + \dots +
 +
\lambda _ {n} x _ {n} ;
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422028.png" />, inequality (3) takes the form
+
for $  \lambda _ {1} = \dots = \lambda _ {n} = 1/n $,  
 +
inequality (3) takes 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/j/j054/j054220/j05422029.png" /></td> </tr></table>
+
$$
 +
( x _ {1} \dots x _ {n} )  ^ {1/n}  \leq  \
 +
 
 +
\frac{x _ {1} + \dots + x _ {n} }{n}
 +
.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Hölder,  "Ueber einen Mittelwertsatz"  ''Göttinger Nachr.''  (1889)  pp. 38–47</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Jensen,  "Sur les fonctions convexes et les inégualités entre les valeurs moyennes"  ''Acta Math.'' , '''30'''  (1906)  pp. 175–193</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Hölder,  "Ueber einen Mittelwertsatz"  ''Göttinger Nachr.''  (1889)  pp. 38–47</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Jensen,  "Sur les fonctions convexes et les inégualités entre les valeurs moyennes"  ''Acta Math.'' , '''30'''  (1906)  pp. 175–193</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,  J.E. Littlewood,  G. Pólya,  "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Jensen's inequality (2) can be generalized by taking instead a [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422030.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422031.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422032.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422034.png" /> a bounded real-valued function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422036.png" /> a convex function on the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054220/j05422037.png" />; then
+
Jensen's inequality (2) can be generalized by taking instead a [[Probability measure|probability measure]] $  \mu $
 +
on a $  \sigma $-
 +
algebra $  {\mathcal M} $
 +
in a set $  D \subset  \mathbf R $,  
 +
$  x $
 +
a bounded real-valued function in $  L _ {1} ( \mu ) $
 +
and $  f $
 +
a convex function on the range of $  x $;  
 +
then
  
<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/j/j054/j054220/j05422038.png" /></td> </tr></table>
+
$$
 +
f \left ( \int\limits _ { D } x  d \mu \right )  \leq  \
 +
\int\limits _ { D } ( f \circ x)  d \mu .
 +
$$
  
 
For another generalization cf. [[#References|[a2]]].
 
For another generalization cf. [[#References|[a2]]].

Latest revision as of 22:14, 5 June 2020


in the simplest discrete form

The inequality

$$ \tag{1 } f ( \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} ) \leq \ \lambda _ {1} f ( x _ {1} ) + \dots + \lambda _ {n} f ( x _ {n} ), $$

where $ f $ is a convex function on some set $ C $ in $ \mathbf R $( see Convex function (of a real variable)), $ x _ {i} \in C $, $ \lambda _ {i} \geq 0 $, $ i = 1 \dots n $, and

$$ \lambda _ {1} + \dots + \lambda _ {n} = 1. $$

Equality holds if and only if $ x _ {1} = \dots = x _ {n} $ or if $ f $ is linear. Jensen's integral inequality for a convex function $ f $ is:

$$ \tag{2 } f \left ( \int\limits _ { D } \lambda ( t) x ( t) dt \right ) \leq \int\limits _ { D } \lambda ( t) f ( x ( t)) dt, $$

where $ x ( D) \subset C $, $ \lambda ( t) \geq 0 $ for $ t \in D $ and

$$ \int\limits _ { D } \lambda ( t) dt = 1. $$

Equality holds if and only if either $ x ( t) = \textrm{ const } $ on $ D $ or if $ f $ is linear on $ x ( D) $. If $ f $ is a concave function, the inequality signs in (1) and (2) must be reversed. Inequality (1) was established by O. Hölder, and (2) by J.L. Jensen [2].

With suitable choices of the convex function $ f $ and the weights $ \lambda _ {i} $ or weight function $ \lambda $, inequalities (1) and (2) become concrete inequalities, among which one finds the majority of the classical inequalities. For example, if in (1) one sets $ f( x) = - \mathop{\rm ln} x $, $ x > 0 $, then one obtains an inequality between the weighted arithmetic mean and the geometric mean:

$$ \tag{3 } x _ {1} ^ {\lambda _ {1} } \dots x _ {n} ^ {\lambda _ {n} } \leq \ \lambda _ {1} x _ {1} + \dots + \lambda _ {n} x _ {n} ; $$

for $ \lambda _ {1} = \dots = \lambda _ {n} = 1/n $, inequality (3) takes the form

$$ ( x _ {1} \dots x _ {n} ) ^ {1/n} \leq \ \frac{x _ {1} + \dots + x _ {n} }{n} . $$

References

[1] O. Hölder, "Ueber einen Mittelwertsatz" Göttinger Nachr. (1889) pp. 38–47
[2] J.L. Jensen, "Sur les fonctions convexes et les inégualités entre les valeurs moyennes" Acta Math. , 30 (1906) pp. 175–193
[3] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)

Comments

Jensen's inequality (2) can be generalized by taking instead a probability measure $ \mu $ on a $ \sigma $- algebra $ {\mathcal M} $ in a set $ D \subset \mathbf R $, $ x $ a bounded real-valued function in $ L _ {1} ( \mu ) $ and $ f $ a convex function on the range of $ x $; then

$$ f \left ( \int\limits _ { D } x d \mu \right ) \leq \ \int\limits _ { D } ( f \circ x) d \mu . $$

For another generalization cf. [a2].

References

[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1978) pp. 24
[a2] P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1988) pp. 27ff
How to Cite This Entry:
Jensen inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jensen_inequality&oldid=47465
This article was adapted from an original article by E.K. Godunova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article