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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909001.png" /> of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909002.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909004.png" />, defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909005.png" /> and possessing the following properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909006.png" /> is upper semi-continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909007.png" />; 2) for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909008.png" /> there exist values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s0909009.png" />, arbitrarily small, such that
+
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$#C+1 = 225 : ~/encyclopedia/old_files/data/S090/S.0900900 Subharmonic function
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<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/s/s090/s090900/s09090010.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090011.png" /> is the mean value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090012.png" /> over the area of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090013.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090014.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090016.png" /> is the area of the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090017.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090018.png" /> (this condition is sometimes dropped). In this definition of a subharmonic function, the mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090019.png" /> over the area of the sphere can be replaced by the mean value
+
A function $  u = u( x): D \rightarrow [ - \infty , \infty ) $
 +
of the points  $  x = ( x _ {1} \dots x _ {n} ) $
 +
of a Euclidean space  $  \mathbf R  ^ {n} $,
 +
$  n \geq  2 $,
 +
defined in a domain  $  D \subset  \mathbf R  ^ {n} $
 +
and possessing the following properties: 1)  $  u( x) $
 +
is upper semi-continuous in $  D $;  
 +
2) for any point  $  x _ {0} \in D $
 +
there exist values  $  r > 0 $,
 +
arbitrarily small, such that
  
<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/s/s090/s090900/s09090020.png" /></td> </tr></table>
+
$$
 +
u( x _ {0} )  \leq  I( u; x _ {0} , r)  = \
  
over the volume of the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090022.png" /> is the volume of the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090023.png" />.
+
\frac{1}{s _ {n} r  ^ {n-} 1 }
 +
\int\limits _ {S( x _ {0} ,r) } u( x)  d \sigma ( x),
 +
$$
  
An equivalent definition of a subharmonic function, which explains the name "subharmonic function" is obtained by replacing condition 2) by 2'): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090024.png" /> is a relatively-compact subdomain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090026.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090027.png" /> that is continuous on the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090028.png" /> and is such that
+
where  $  I( u;  x _ {0} , r) $
 +
is the mean value of the function $  u( x) $
 +
over the area of the sphere $ S( x _ {0} , r) $
 +
with centre  $  x _ {0} $
 +
of radius  $  r $
 +
and  $  s _ {n} = 2 \pi  ^ {n/2} \Gamma ( n/2) $
 +
is the area of the unit sphere in  $  \mathbf R  ^ {n} $;
 +
and 3)  $  u( x) \not\equiv - \infty $(
 +
this condition is sometimes dropped). In this definition of a subharmonic function, the mean value  $  I( u;  x _ {0} , r) $
 +
over the area of the sphere can be replaced by the mean value
  
<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/s/s090/s090900/s09090029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$
 +
J( u; x _ {0} , r)  =
 +
\frac{1}{v _ {n} r  ^ {n} }
 +
\int\limits _ {B( x _ {0} ,r) }
 +
u( x)  dv( x)
 +
$$
  
on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090030.png" />, then the inequality (1) holds everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090031.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090032.png" /> is called a harmonic majorant of the subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090034.png" />). If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090035.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090036.png" />, then for it to be subharmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090037.png" /> it is necessary and sufficient that the result of applying the [[Laplace operator|Laplace operator]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090038.png" />, be non-negative in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090039.png" />.
+
over the volume of the ball  $  B( x _ {0} , r) $,  
 +
where  $  v _ {n} = s _ {n} /n $
 +
is the volume of the unit ball in $  \mathbf R  ^ {n} $.
  
The idea of a subharmonic function was expounded in essence by H. Poincaré in the [[Balayage method|balayage method]]. Subharmonic functions are also found in the work of F. Hartogs [[#References|[1]]] on the theory of analytic functions of several complex variables; the systematic study of subharmonic functions began with the work of F. Riesz . The close connection between subharmonic functions and analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090040.png" /> of one or several complex variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090042.png" />, and the consequent possible use of subharmonic functions for the study of analytic functions is related to the fact that the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090043.png" /> and the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090044.png" /> of the modulus of an analytic function are subharmonic functions. On the other hand, condition 2') shows that subharmonic functions can be considered as the analogue of convex functions of one real variable (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
+
An equivalent definition of a subharmonic function, which explains the name  "subharmonic function"  is obtained by replacing condition 2) by 2'): If  $  \Delta $
 +
is a relatively-compact subdomain of  $  D $
 +
and  $  v( x) $
 +
is a harmonic function in  $  \Delta $
 +
that is continuous on the closure  $  \overline \Delta \; $
 +
and is such that
 +
 
 +
$$ \tag{1 }
 +
u( x)  \leq  v( x)
 +
$$
 +
 
 +
on the boundary  $  \partial  \Delta $,
 +
then the inequality (1) holds everywhere in  $  \Delta $(
 +
$  v( x) $
 +
is called a harmonic majorant of the subharmonic function  $  u( x) $
 +
in  $  \Delta $).
 +
If the function  $  u( x) $
 +
belongs to the class  $  C  ^ {2} ( D) $,
 +
then for it to be subharmonic in  $  D $
 +
it is necessary and sufficient that the result of applying the [[Laplace operator|Laplace operator]],  $  \Delta u $,
 +
be non-negative in  $  D $.
 +
 
 +
The idea of a subharmonic function was expounded in essence by H. Poincaré in the [[Balayage method|balayage method]]. Subharmonic functions are also found in the work of F. Hartogs [[#References|[1]]] on the theory of analytic functions of several complex variables; the systematic study of subharmonic functions began with the work of F. Riesz . The close connection between subharmonic functions and analytic functions $  f( z) $
 +
of one or several complex variables $  z = ( z _ {1} \dots z _ {n} ) $,  
 +
$  n \geq  1 $,  
 +
and the consequent possible use of subharmonic functions for the study of analytic functions is related to the fact that the modulus $  | f( z) | $
 +
and the logarithm $  \mathop{\rm ln}  | f( z) | $
 +
of the modulus of an analytic function are subharmonic functions. On the other hand, condition 2') shows that subharmonic functions can be considered as the analogue of convex functions of one real variable (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
  
 
Simple properties of subharmonic functions.
 
Simple properties of subharmonic functions.
  
1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090045.png" /> are subharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090047.png" /> are non-negative numbers, then the linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090048.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090049.png" />.
+
1) If $  u _ {1} \dots u _ {m} $
 +
are subharmonic functions in $  D $
 +
and $  \lambda _ {1} \dots \lambda _ {m} $
 +
are non-negative numbers, then the linear combination $  \sum _ {k=} 1  ^ {m} \lambda _ {k} u _ {k} $
 +
is a subharmonic function in $  D $.
  
2) The upper envelope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090050.png" /> of a finite family of subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090051.png" /> is a subharmonic function. If the upper envelope of an infinite family of subharmonic functions is upper semi-continuous, then it is also a subharmonic function.
+
2) The upper envelope $  \sup \{ {u _ {k} ( x) } : {1 \leq  k \leq  m } \} $
 +
of a finite family of subharmonic functions $  \{ u _ {k} \} _ {k=} 1  ^ {m} $
 +
is a subharmonic function. If the upper envelope of an infinite family of subharmonic functions is upper semi-continuous, then it is also a subharmonic function.
  
 
3) Uniformly-converging and monotone-decreasing sequences of subharmonic functions converge to subharmonic functions.
 
3) Uniformly-converging and monotone-decreasing sequences of subharmonic functions converge to subharmonic functions.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090052.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090054.png" /> is a convex non-decreasing function on the domain of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090055.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090057.png" />, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090058.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090060.png" /> is a convex function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090061.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090062.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090063.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090064.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090067.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090070.png" />, are subharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090071.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090072.png" /> is a harmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090073.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090075.png" />, is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090076.png" />.
+
4) If $  u( x) $
 +
is a subharmonic function in $  D $
 +
and $  \phi ( u) $
 +
is a convex non-decreasing function on the domain of values $  E $
 +
of the function $  u $
 +
in $  D $,  
 +
or if $  u( x) $
 +
is a harmonic function in $  D $
 +
and $  \phi ( u) $
 +
is a convex function on $  E $,  
 +
then $  \phi ( u( x)) $
 +
is a subharmonic function in $  D $.  
 +
In particular, if $  u( x) $
 +
is a subharmonic function in $  D $,  
 +
then $  e ^ {\lambda u( x) } $,
 +
$  \lambda > 0 $,  
 +
and $  [ u  ^ {+} ( x)]  ^ {k} $,  
 +
$  k \geq  1 $,  
 +
where $  u  ^ {+} ( x) = \max \{ u( x), 0 \} $,  
 +
are subharmonic functions in $  D $;  
 +
if $  u( x) $
 +
is a harmonic function in $  D $,  
 +
then $  | u( x) |  ^ {k} $,  
 +
$  k \geq  1 $,  
 +
is a subharmonic function in $  D $.
  
5) The maximum principle: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090077.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090078.png" /> and for any boundary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090079.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090080.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090082.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090083.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090084.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090086.png" />. This property also holds for unbounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090087.png" />, where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090088.png" /> the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090089.png" /> is taken to mean the exterior of a sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090090.png" />.
+
5) The maximum principle: If $  u( x) $
 +
is a subharmonic function in $  D $
 +
and for any boundary point $  \xi \in \partial  D $
 +
and any $  \epsilon > 0 $
 +
there is a neighbourhood $  V = V( \xi ) $
 +
such that $  u( x) < \epsilon $
 +
in $  D \cap V $,  
 +
then either $  u( x) < 0 $
 +
or $  u( x) \equiv 0 $
 +
in $  D $.  
 +
This property also holds for unbounded domains $  D $,  
 +
where for $  \xi = \infty \in \partial  D $
 +
the neighbourhood $  V $
 +
is taken to mean the exterior of a sphere, $  V = V( \infty ) = \{ {x \in \mathbf R  ^ {n} } : {| x | > R } \} $.
  
6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090091.png" /> is a subharmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090092.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090094.png" /> is a holomorphic mapping of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090095.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090096.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090097.png" /> is a subharmonic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090098.png" />.
+
6) If $  u( x) $
 +
is a subharmonic function in a domain $  D $
 +
of the complex plane $  \mathbf C $
 +
and $  z = z( w) $
 +
is a holomorphic mapping of a domain $  D  ^  \prime  \subset  \mathbf C  ^ {n} $
 +
into $  D $,  
 +
then $  u( z( w)) $
 +
is a subharmonic function in $  D  ^  \prime  $.
  
7) A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s09090099.png" /> is harmonic in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900100.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900102.png" /> are subharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900103.png" /> (cf. [[Harmonic function|Harmonic function]]).
+
7) A function $  u( x) $
 +
is harmonic in the domain $  D \subset  \mathbf R  ^ {n} $
 +
if and only if $  u( x) $
 +
and $  - u( x) $
 +
are subharmonic functions in $  D $(
 +
cf. [[Harmonic function|Harmonic function]]).
  
8) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900104.png" /> is a subharmonic function on the whole plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900105.png" /> that is bounded above, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900106.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900107.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900108.png" /> this property does not hold).
+
8) If $  u( x) $
 +
is a subharmonic function on the whole plane $  \mathbf R  ^ {2} $
 +
that is bounded above, then $  u( x) = \textrm{ const } $(
 +
in $  \mathbf R  ^ {n} $
 +
when $  n \geq  3 $
 +
this property does not hold).
  
 
The [[Perron method|Perron method]] for solving the [[Dirichlet problem|Dirichlet problem]] for harmonic functions is based on the properties 2), 5) and 7).
 
The [[Perron method|Perron method]] for solving the [[Dirichlet problem|Dirichlet problem]] for harmonic functions is based on the properties 2), 5) and 7).
  
The convexity properties of the mean values of subharmonic functions are of great importance: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900109.png" /> is a subharmonic function in an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900111.png" />, then the mean values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900113.png" />, as well as the maximum
+
The convexity properties of the mean values of subharmonic functions are of great importance: If $  u( x) $
 +
is a subharmonic function in an annulus $  r _ {1} \leq  r = | x - x _ {0} | \leq  r _ {2} $,  
 +
$  0 < r _ {1} < r _ {2} $,  
 +
then the mean values $  I( u;  x _ {0} , r) $,  
 +
$  J( u;  x _ {0} , r) $,  
 +
as well as the maximum
  
<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/s/s090/s090900/s090900114.png" /></td> </tr></table>
+
$$
 +
M( u; x _ {0} , r)  = \max \{ {u( x ) } : {| x- x _ {0} | = r } \}
 +
,
 +
$$
  
are convex functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900115.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900116.png" />, or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900117.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900118.png" />, on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900119.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900120.png" /> is a subharmonic function in the disc (ball) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900121.png" />, then, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900123.png" /> are continuous non-decreasing functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900124.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900125.png" />; moreover
+
are convex functions in $  \mathop{\rm ln}  r $
 +
for $  n= 2 $,  
 +
or in $  r  ^ {2-} n $
 +
for $  n \geq  3 $,  
 +
on the interval $  r _ {1} \leq  r \leq  r _ {2} $;  
 +
if $  u( x) $
 +
is a subharmonic function in the disc (ball) $  0 \leq  r = | x- x _ {0} | \leq  r _ {2} $,  
 +
then, moreover, $  I( u;  x _ {0} , r) $
 +
and $  J( u;  x _ {0} , r) $
 +
are continuous non-decreasing functions in $  r $
 +
on the interval 0 \leq  r \leq  r _ {2} $;  
 +
moreover
  
<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/s/s090/s090900/s090900126.png" /></td> </tr></table>
+
$$
 +
I( u; x _ {0} , 0)  = J( u; x _ {0} , 0)  = u( x _ {0} );
 +
$$
  
 
in the latter instance,
 
in the latter instance,
  
<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/s/s090/s090900/s090900127.png" /></td> </tr></table>
+
$$
 +
u( x _ {0} )  \leq  J( u; x _ {0} , r)  \leq  I( u; x _ {0} , r)
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900128.png" />. The mean values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900130.png" />, considered as functions of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900131.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900133.png" />, are subharmonic functions in the corresponding subdomain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900134.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900135.png" /> is continuous. By forming iterations of a sufficiently-high order,
+
for 0 \leq  r \leq  r _ {2} $.  
 +
The mean values $  I( u;  x _ {0} , r) $
 +
and $  J( u;  x _ {0} , r) $,  
 +
considered as functions of the point $  x _ {0} $
 +
for fixed $  u $
 +
and $  r $,  
 +
are subharmonic functions in the corresponding subdomain $  D  ^  \prime  \subset  D $,  
 +
and $  J( u;  x _ {0} , r) $
 +
is continuous. By forming iterations of a sufficiently-high order,
  
<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/s/s090/s090900/s090900136.png" /></td> </tr></table>
+
$$
 +
u _ {m}  ^ {(} k) ( x)  = J \left ( u _ {m}  ^ {(} k- 1) ; x,
 +
\frac{1}{m}
 +
\right ) ,\ \
 +
u _ {m}  ^ {(} 0) ( x)  = u( x),
 +
$$
  
it is possible to obtain a monotone-decreasing sequence of subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900137.png" /> of any degree of smoothness that converges, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900138.png" />, to an arbitrarily given subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900139.png" />.
+
it is possible to obtain a monotone-decreasing sequence of subharmonic functions $  \{ u _ {m}  ^ {(} k) ( x) \} _ {m= m _ {0}  }  ^  \infty  $
 +
of any degree of smoothness that converges, as $  m \rightarrow \infty $,  
 +
to an arbitrarily given subharmonic function $  u( x) $.
  
The [[Newton potential|Newton potential]] and [[Logarithmic potential|logarithmic potential]] of non-negative masses, when written with a minus sign, are subharmonic functions everywhere in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900140.png" />. On the other hand, one of the basic theorems in the theory of subharmonic functions is the Riesz local representation theorem: An arbitrary subharmonic function can be represented as the sum of a harmonic function and a potential with a minus sign (see ). More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900141.png" /> is a subharmonic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900142.png" />, then there exists a unique non-negative Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900143.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900144.png" /> (a measure associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900145.png" />, or a Riesz measure) such that for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900146.png" /> the representation
+
The [[Newton potential|Newton potential]] and [[Logarithmic potential|logarithmic potential]] of non-negative masses, when written with a minus sign, are subharmonic functions everywhere in the space $  \mathbf R  ^ {n} $.  
 +
On the other hand, one of the basic theorems in the theory of subharmonic functions is the Riesz local representation theorem: An arbitrary subharmonic function can be represented as the sum of a harmonic function and a potential with a minus sign (see ). More precisely, if $  u( x) $
 +
is a subharmonic function in a domain $  D \subset  \mathbf R  ^ {n} $,  
 +
then there exists a unique non-negative Borel measure $  \mu $
 +
on $  D $(
 +
a measure associated with $  u( x) $,  
 +
or a Riesz measure) such that for any compact set $  E \subset  D $
 +
the representation
  
<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/s/s090/s090900/s090900147.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
u( x)  = \int\limits _ { E } K( x- \xi )  d \mu ( \xi ) + h( x)
 +
$$
  
is valid, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900148.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900149.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900150.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900151.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900152.png" /> is a harmonic function in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900153.png" />. The Riesz theorem establishes a close link between the theory of subharmonic functions and [[Potential theory|potential theory]].
+
is valid, where $  K( x- \xi ) = \mathop{\rm ln}  | x- \xi | $
 +
when $  n= 2 $,  
 +
$  K( x- \xi ) = - |  x- \xi |  ^ {2-} n $
 +
when $  n \geq  3 $,  
 +
and where $  h( x) $
 +
is a harmonic function in the interior of $  E $.  
 +
The Riesz theorem establishes a close link between the theory of subharmonic functions and [[Potential theory|potential theory]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900154.png" /> is a regular closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900155.png" /> bounded, for example, by a Lyapunov surface, and having a [[Green function|Green function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900156.png" />, then as well as (2) a representation using the Green potential is valid:
+
If $  E $
 +
is a regular closed domain $  \overline{G}\; $
 +
bounded, for example, by a Lyapunov surface, and having a [[Green function|Green function]] $  g( x, \xi ) $,
 +
then as well as (2) a representation using the Green potential is valid:
  
<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/s/s090/s090900/s090900157.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
u( x)  = - \int\limits _ {\overline{G}\; } g( x, \xi )  d \mu ( \xi ) + w( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900158.png" /> is the least harmonic majorant of the subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900159.png" /> in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900160.png" />.
+
where $  w( x) $
 +
is the least harmonic majorant of the subharmonic function $  u( x) $
 +
in the domain $  G $.
  
A representation in the form (3), generally speaking, does not hold in the whole domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900161.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900162.png" />, and in the theory of subharmonic functions great importance is attached to the question of distinguishing the class of subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900163.png" /> that allow a representation (3) in the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900164.png" />, i.e. the question of distinguishing the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900165.png" /> of subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900166.png" /> that have a harmonic majorant in the whole domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900167.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900168.png" /> is a ball (disc) and there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900169.png" /> such that
+
A representation in the form (3), generally speaking, does not hold in the whole domain of definition $  D $
 +
of $  u( x) $,  
 +
and in the theory of subharmonic functions great importance is attached to the question of distinguishing the class of subharmonic functions $  u( x) $
 +
that allow a representation (3) in the whole domain $  D $,  
 +
i.e. the question of distinguishing the class $  A $
 +
of subharmonic functions $  u( x) $
 +
that have a harmonic majorant in the whole domain $  D $.  
 +
For example, if $  D = B( 0, R) $
 +
is a ball (disc) and there exists a constant $  C = C( u) $
 +
such that
  
<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/s/s090/s090900/s090900170.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\int\limits _ {S( 0,R) } u  ^ {+} ( r \xi )  d \sigma ( \xi )  < C( u) < + \infty ,\ \
 +
0 < r < 1 ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900171.png" /> allows a representation (3) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900172.png" />, and the least harmonic majorant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900173.png" /> is, in turn, represented by a Poisson–Stieltjes integral:
+
then $  u( x) $
 +
allows a representation (3) in $  D $,  
 +
and the least harmonic majorant $  w( x) $
 +
is, in turn, represented by a Poisson–Stieltjes integral:
  
<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/s/s090/s090900/s090900174.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
w( x)  = \int\limits
 +
\frac{R  ^ {n-} 2 ( R  ^ {2} - | x |  ^ {2} ) }{| x- \xi |  ^ {n} }
 +
  d
 +
\nu ( \xi ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900175.png" /> is a Borel measure of arbitrary sign concentrated on the boundary sphere (circle) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900176.png" /> (a boundary measure) and normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900177.png" />.
+
where $  \nu $
 +
is a Borel measure of arbitrary sign concentrated on the boundary sphere (circle) $  \partial  D = S( 0, R) $(
 +
a boundary measure) and normalized by the condition $  \int d \nu ( \xi ) = 1 $.
  
With regard to (5), it is important in practical applications to know under which conditions the boundary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900178.png" /> has only a negative singular component, i.e. under which conditions the component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900179.png" /> in the canonical decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900180.png" /> is absolutely continuous. This question is answered by introducing the class of strictly-subharmonic functions (see [[#References|[11]]]–[[#References|[13]]], [[#References|[15]]], as well as [[#References|[10]]], where generalizations are also examined). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900181.png" /> be an increasing concave function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900182.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900183.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900184.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900185.png" />, is said to be strictly subharmonic relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900186.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900187.png" /> is a subharmonic function. For example, logarithmically-subharmonic functions (cf. [[Logarithmically-subharmonic function|Logarithmically-subharmonic function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900188.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900189.png" /> is a subharmonic function, belong to the class of strictly-subharmonic functions. If condition (4) is fulfilled for a strictly-subharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900190.png" /> in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900191.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900192.png" /> can be represented in the form (3) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900193.png" />, and the boundary measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900194.png" /> is characterized by the decomposition
+
With regard to (5), it is important in practical applications to know under which conditions the boundary measure $  \nu $
 +
has only a negative singular component, i.e. under which conditions the component $  \nu  ^ {+} $
 +
in the canonical decomposition $  \nu = \nu  ^ {+} - \nu  ^ {-} $
 +
is absolutely continuous. This question is answered by introducing the class of strictly-subharmonic functions (see [[#References|[11]]]–[[#References|[13]]], [[#References|[15]]], as well as [[#References|[10]]], where generalizations are also examined). Let $  \psi ( y) $
 +
be an increasing concave function in $  y $
 +
for which $  \lim\limits _ {y \rightarrow + \infty }  y/ \psi ( y) = + \infty $.  
 +
A function $  u( x) $,  
 +
$  x \in D $,  
 +
is said to be strictly subharmonic relative to $  \psi ( y) $
 +
if $  \psi ( u( x)) $
 +
is a subharmonic function. For example, logarithmically-subharmonic functions (cf. [[Logarithmically-subharmonic function|Logarithmically-subharmonic function]]) $  u( x) \geq  0 $,  
 +
for which $  \mathop{\rm ln}  u( x) $
 +
is a subharmonic function, belong to the class of strictly-subharmonic functions. If condition (4) is fulfilled for a strictly-subharmonic function $  u( x) $
 +
in the ball $  D $,  
 +
then $  u( x) $
 +
can be represented in the form (3) in $  D $,  
 +
and the boundary measure $  \nu $
 +
is characterized by the decomposition
  
<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/s/s090/s090900/s090900195.png" /></td> </tr></table>
+
$$
 +
d \nu ( \xi ) = u( \xi )  d \sigma ( \xi ) - d \nu  ^ {-} ( \xi ) ,\ \
 +
\xi \in \partial  D,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900196.png" /> are the radial boundary values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900197.png" /> (which exist almost everywhere with respect to the Lebesgue measure on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900198.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900199.png" /> is the singular component of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900200.png" />.
+
where $  u( \xi ) $
 +
are the radial boundary values of the function $  u( x) $(
 +
which exist almost everywhere with respect to the Lebesgue measure on the sphere $  \partial  D = S( 0, R) $),  
 +
and $  \nu  ^ {-} \geq  0 $
 +
is the singular component of the measure $  \nu $.
  
Subharmonic functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900201.png" /> in the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900202.png" /> have radial boundary values almost everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900203.png" />. However, examples have been constructed of bounded, continuous subharmonic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900204.png" /> that do not have non-tangential boundary values anywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900205.png" />, a phenomenon that does not occur for harmonic functions. For non-tangential boundary values to exist, apart from (4) further conditions have to be imposed on the associated measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900206.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900207.png" /> (see, for example, [[#References|[14]]]).
+
Subharmonic functions of class $  A $
 +
in the ball $  D $
 +
have radial boundary values almost everywhere on $  \partial  D = S( 0, R) $.  
 +
However, examples have been constructed of bounded, continuous subharmonic functions in $  D $
 +
that do not have non-tangential boundary values anywhere on $  \partial  D $,  
 +
a phenomenon that does not occur for harmonic functions. For non-tangential boundary values to exist, apart from (4) further conditions have to be imposed on the associated measure $  \mu $
 +
in $  D $(
 +
see, for example, [[#References|[14]]]).
  
One of the essential questions in the theory of subharmonic functions and its applications is the characterization of the boundary properties of functions of different subclasses of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900208.png" />. The general method of introducing these subclasses consists of the fact that for strictly-subharmonic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900209.png" /> relative to a concave function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900210.png" />, any non-decreasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900211.png" /> is examined that is convex relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900212.png" /> and that is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900213.png" />, and the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900214.png" /> is introduced. For a sphere it is defined by the condition
+
One of the essential questions in the theory of subharmonic functions and its applications is the characterization of the boundary properties of functions of different subclasses of the class $  A $.  
 +
The general method of introducing these subclasses consists of the fact that for strictly-subharmonic functions $  u( x) $
 +
relative to a concave function $  \psi ( y) $,  
 +
any non-decreasing function $  \phi ( y) $
 +
is examined that is convex relative to $  \psi ( y) $
 +
and that is such that $  \lim\limits _ {y \rightarrow + \infty }  \phi ( y)/ \psi ( y) = + \infty $,  
 +
and the class $  A _  \phi  $
 +
is introduced. For a sphere it is defined by the condition
  
<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/s/s090/s090900/s090900215.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {S( 0,R) }
 +
\phi  ^ {+} ( u( r \xi ))  d \sigma ( \xi )  < C( u, \phi )  < + \infty ,\ \
 +
0 < r < 1.
 +
$$
  
 
For the boundary properties of subharmonic functions, see [[#References|[3]]]–[[#References|[5]]], [[#References|[10]]]–[[#References|[16]]].
 
For the boundary properties of subharmonic functions, see [[#References|[3]]]–[[#References|[5]]], [[#References|[10]]]–[[#References|[16]]].
Line 93: Line 340:
 
For functions that can be represented as the difference between two subharmonic functions, the concept of characteristic in the sense of R. Nevanlinna has been introduced, and the theory of functions of bounded characteristic (cf. [[Function of bounded characteristic|Function of bounded characteristic]]) has been generalized (see [[#References|[3]]], ).
 
For functions that can be represented as the difference between two subharmonic functions, the concept of characteristic in the sense of R. Nevanlinna has been introduced, and the theory of functions of bounded characteristic (cf. [[Function of bounded characteristic|Function of bounded characteristic]]) has been generalized (see [[#References|[3]]], ).
  
A distinctive generalization of the theory of entire functions (cf. [[Entire function|Entire function]]) is the theory of subharmonic functions in the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900216.png" />. Here, generalizations of the Weierstrass and Hadamard classical representation theorems of entire functions have been obtained, along with the theory of the growth and value distribution, the theory of asymptotic values and asymptotic paths, etc. (see ).
+
A distinctive generalization of the theory of entire functions (cf. [[Entire function|Entire function]]) is the theory of subharmonic functions in the whole space $  \mathbf R  ^ {n} $.  
 +
Here, generalizations of the Weierstrass and Hadamard classical representation theorems of entire functions have been obtained, along with the theory of the growth and value distribution, the theory of asymptotic values and asymptotic paths, etc. (see ).
  
 
In the theory of analytic functions of several complex variables, the study of the subclasses of plurisubharmonic functions and pluriharmonic functions (cf. [[Plurisubharmonic function|Plurisubharmonic function]]; [[Pluriharmonic function|Pluriharmonic function]]) is of considerable importance (see [[#References|[17]]]). For axiomatic generalizations of subharmonic functions, see [[#References|[9]]].
 
In the theory of analytic functions of several complex variables, the study of the subclasses of plurisubharmonic functions and pluriharmonic functions (cf. [[Plurisubharmonic function|Plurisubharmonic function]]; [[Pluriharmonic function|Pluriharmonic function]]) is of considerable importance (see [[#References|[17]]]). For axiomatic generalizations of subharmonic functions, see [[#References|[9]]].
Line 99: Line 347:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hartogs,  "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen welche nach Potenzen einer Veränderlichen fortschreiten"  ''Math. Ann.'' , '''62'''  (1906)  pp. 1–88</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Privalov,  "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. Privalov,  "Boundary properties of single-valued analytic functions" , Moscow  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T. Radó,  "Subharmonic functions" , Springer  (1937)</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  W.K. Hayman,  "Subharmonic functions" , '''2''' , Acad. Press  (1989)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M. Brelot,  "Etude des fonctions sousharmoniques au voisinage d'un point" , Hermann  (1934)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1967)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Heins,  "Hardy classes on Riemann surfaces" , Springer  (1969)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  E.D. Solomentsev,  "On some classes of subharmonic functions"  ''Izv. Akad. Nauk SSSR'' , '''5–6'''  (1938)  pp. 571–582  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  I.I. Privalov,  P.I. Kuznetsov,  "Sur les problèmes limites et les classes différentes de fonctions harmoniques et sousharmoniques définies dans une domaine arbitraire"  ''Mat. Sb.'' , '''6''' :  3  (1939)  pp. 345–376  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  E.D. Solomentsev,  "Classes of functions subharmonic on a half-space"  ''Vestnik Moskov. Gos. Univ. Ser. Mat.-Mekh. Astron.'' :  5  (1959)  pp. 73–91  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  E.D. Solomentsev,  "On boundary values of subharmonic functions"  ''Czech. Math. J.'' , '''8''' :  4  (1958)  pp. 520–536  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  L. Gårding,  L. Hormander,  "Strongly subharmonic functions"  ''Math. Scand.'' , '''15'''  (1964)  pp. 93–96</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  E.D. Solomentsev,  "Harmonic and subharmonic functions and their generalization"  ''Itogi Nauk Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962''  (1964)  pp. 83–100  (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Hartogs,  "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen welche nach Potenzen einer Veränderlichen fortschreiten"  ''Math. Ann.'' , '''62'''  (1906)  pp. 1–88</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. Privalov,  "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. Privalov,  "Boundary properties of single-valued analytic functions" , Moscow  (1941)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T. Radó,  "Subharmonic functions" , Springer  (1937)</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''1''' , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  W.K. Hayman,  "Subharmonic functions" , '''2''' , Acad. Press  (1989)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M. Brelot,  "Etude des fonctions sousharmoniques au voisinage d'un point" , Hermann  (1934)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Brélot,  "Lectures on potential theory" , Tata Inst.  (1967)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Heins,  "Hardy classes on Riemann surfaces" , Springer  (1969)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  E.D. Solomentsev,  "On some classes of subharmonic functions"  ''Izv. Akad. Nauk SSSR'' , '''5–6'''  (1938)  pp. 571–582  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  I.I. Privalov,  P.I. Kuznetsov,  "Sur les problèmes limites et les classes différentes de fonctions harmoniques et sousharmoniques définies dans une domaine arbitraire"  ''Mat. Sb.'' , '''6''' :  3  (1939)  pp. 345–376  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  E.D. Solomentsev,  "Classes of functions subharmonic on a half-space"  ''Vestnik Moskov. Gos. Univ. Ser. Mat.-Mekh. Astron.'' :  5  (1959)  pp. 73–91  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  E.D. Solomentsev,  "On boundary values of subharmonic functions"  ''Czech. Math. J.'' , '''8''' :  4  (1958)  pp. 520–536  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  L. Gårding,  L. Hormander,  "Strongly subharmonic functions"  ''Math. Scand.'' , '''15'''  (1964)  pp. 93–96</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  E.D. Solomentsev,  "Harmonic and subharmonic functions and their generalization"  ''Itogi Nauk Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962''  (1964)  pp. 83–100  (In Russian)</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Axiomatic potential theory can be founded upon the properties 1) and 3), completed by some additional properties of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900217.png" /> of negative subharmonic functions: a) the upper semi-continuous regularization of the supremum of a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900218.png" /> is subharmonic; b) (the Riesz decomposition property) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900219.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900220.png" />, there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900221.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900222.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900223.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900224.png" />; and c) any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090900/s090900225.png" /> is the limit of a decreasing sequence in a sufficiently nice subcone of continuous subharmonic functions. See [[#References|[a1]]].
+
Axiomatic potential theory can be founded upon the properties 1) and 3), completed by some additional properties of the set $  S $
 +
of negative subharmonic functions: a) the upper semi-continuous regularization of the supremum of a subset of $  S $
 +
is subharmonic; b) (the Riesz decomposition property) for any $  u , v, w \in S $,  
 +
$  u \geq  v+ w $,  
 +
there exist $  v  ^  \prime  , w  ^  \prime  \in S $
 +
such that $  u= v  ^  \prime  + w  ^  \prime  $,  
 +
$  v  ^  \prime  \geq  v $,  
 +
$  w  ^  \prime  \geq  w $;  
 +
and c) any $  u \in S $
 +
is the limit of a decreasing sequence in a sufficiently nice subcone of continuous subharmonic functions. See [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , Dover, reprint  (1954)  pp. 315ff  (Re-issue: Springer, 1967)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bliedtner,  W. Hansen,  "Potential theory. An analytic and probabilistic approach to balayage" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.I. Ronkin,  "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O.D. Kellogg,  "Foundations of potential theory" , Dover, reprint  (1954)  pp. 315ff  (Re-issue: Springer, 1967)</TD></TR></table>

Revision as of 08:24, 6 June 2020


A function $ u = u( x): D \rightarrow [ - \infty , \infty ) $ of the points $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, defined in a domain $ D \subset \mathbf R ^ {n} $ and possessing the following properties: 1) $ u( x) $ is upper semi-continuous in $ D $; 2) for any point $ x _ {0} \in D $ there exist values $ r > 0 $, arbitrarily small, such that

$$ u( x _ {0} ) \leq I( u; x _ {0} , r) = \ \frac{1}{s _ {n} r ^ {n-} 1 } \int\limits _ {S( x _ {0} ,r) } u( x) d \sigma ( x), $$

where $ I( u; x _ {0} , r) $ is the mean value of the function $ u( x) $ over the area of the sphere $ S( x _ {0} , r) $ with centre $ x _ {0} $ of radius $ r $ and $ s _ {n} = 2 \pi ^ {n/2} \Gamma ( n/2) $ is the area of the unit sphere in $ \mathbf R ^ {n} $; and 3) $ u( x) \not\equiv - \infty $( this condition is sometimes dropped). In this definition of a subharmonic function, the mean value $ I( u; x _ {0} , r) $ over the area of the sphere can be replaced by the mean value

$$ J( u; x _ {0} , r) = \frac{1}{v _ {n} r ^ {n} } \int\limits _ {B( x _ {0} ,r) } u( x) dv( x) $$

over the volume of the ball $ B( x _ {0} , r) $, where $ v _ {n} = s _ {n} /n $ is the volume of the unit ball in $ \mathbf R ^ {n} $.

An equivalent definition of a subharmonic function, which explains the name "subharmonic function" is obtained by replacing condition 2) by 2'): If $ \Delta $ is a relatively-compact subdomain of $ D $ and $ v( x) $ is a harmonic function in $ \Delta $ that is continuous on the closure $ \overline \Delta \; $ and is such that

$$ \tag{1 } u( x) \leq v( x) $$

on the boundary $ \partial \Delta $, then the inequality (1) holds everywhere in $ \Delta $( $ v( x) $ is called a harmonic majorant of the subharmonic function $ u( x) $ in $ \Delta $). If the function $ u( x) $ belongs to the class $ C ^ {2} ( D) $, then for it to be subharmonic in $ D $ it is necessary and sufficient that the result of applying the Laplace operator, $ \Delta u $, be non-negative in $ D $.

The idea of a subharmonic function was expounded in essence by H. Poincaré in the balayage method. Subharmonic functions are also found in the work of F. Hartogs [1] on the theory of analytic functions of several complex variables; the systematic study of subharmonic functions began with the work of F. Riesz . The close connection between subharmonic functions and analytic functions $ f( z) $ of one or several complex variables $ z = ( z _ {1} \dots z _ {n} ) $, $ n \geq 1 $, and the consequent possible use of subharmonic functions for the study of analytic functions is related to the fact that the modulus $ | f( z) | $ and the logarithm $ \mathop{\rm ln} | f( z) | $ of the modulus of an analytic function are subharmonic functions. On the other hand, condition 2') shows that subharmonic functions can be considered as the analogue of convex functions of one real variable (cf. Convex function (of a real variable)).

Simple properties of subharmonic functions.

1) If $ u _ {1} \dots u _ {m} $ are subharmonic functions in $ D $ and $ \lambda _ {1} \dots \lambda _ {m} $ are non-negative numbers, then the linear combination $ \sum _ {k=} 1 ^ {m} \lambda _ {k} u _ {k} $ is a subharmonic function in $ D $.

2) The upper envelope $ \sup \{ {u _ {k} ( x) } : {1 \leq k \leq m } \} $ of a finite family of subharmonic functions $ \{ u _ {k} \} _ {k=} 1 ^ {m} $ is a subharmonic function. If the upper envelope of an infinite family of subharmonic functions is upper semi-continuous, then it is also a subharmonic function.

3) Uniformly-converging and monotone-decreasing sequences of subharmonic functions converge to subharmonic functions.

4) If $ u( x) $ is a subharmonic function in $ D $ and $ \phi ( u) $ is a convex non-decreasing function on the domain of values $ E $ of the function $ u $ in $ D $, or if $ u( x) $ is a harmonic function in $ D $ and $ \phi ( u) $ is a convex function on $ E $, then $ \phi ( u( x)) $ is a subharmonic function in $ D $. In particular, if $ u( x) $ is a subharmonic function in $ D $, then $ e ^ {\lambda u( x) } $, $ \lambda > 0 $, and $ [ u ^ {+} ( x)] ^ {k} $, $ k \geq 1 $, where $ u ^ {+} ( x) = \max \{ u( x), 0 \} $, are subharmonic functions in $ D $; if $ u( x) $ is a harmonic function in $ D $, then $ | u( x) | ^ {k} $, $ k \geq 1 $, is a subharmonic function in $ D $.

5) The maximum principle: If $ u( x) $ is a subharmonic function in $ D $ and for any boundary point $ \xi \in \partial D $ and any $ \epsilon > 0 $ there is a neighbourhood $ V = V( \xi ) $ such that $ u( x) < \epsilon $ in $ D \cap V $, then either $ u( x) < 0 $ or $ u( x) \equiv 0 $ in $ D $. This property also holds for unbounded domains $ D $, where for $ \xi = \infty \in \partial D $ the neighbourhood $ V $ is taken to mean the exterior of a sphere, $ V = V( \infty ) = \{ {x \in \mathbf R ^ {n} } : {| x | > R } \} $.

6) If $ u( x) $ is a subharmonic function in a domain $ D $ of the complex plane $ \mathbf C $ and $ z = z( w) $ is a holomorphic mapping of a domain $ D ^ \prime \subset \mathbf C ^ {n} $ into $ D $, then $ u( z( w)) $ is a subharmonic function in $ D ^ \prime $.

7) A function $ u( x) $ is harmonic in the domain $ D \subset \mathbf R ^ {n} $ if and only if $ u( x) $ and $ - u( x) $ are subharmonic functions in $ D $( cf. Harmonic function).

8) If $ u( x) $ is a subharmonic function on the whole plane $ \mathbf R ^ {2} $ that is bounded above, then $ u( x) = \textrm{ const } $( in $ \mathbf R ^ {n} $ when $ n \geq 3 $ this property does not hold).

The Perron method for solving the Dirichlet problem for harmonic functions is based on the properties 2), 5) and 7).

The convexity properties of the mean values of subharmonic functions are of great importance: If $ u( x) $ is a subharmonic function in an annulus $ r _ {1} \leq r = | x - x _ {0} | \leq r _ {2} $, $ 0 < r _ {1} < r _ {2} $, then the mean values $ I( u; x _ {0} , r) $, $ J( u; x _ {0} , r) $, as well as the maximum

$$ M( u; x _ {0} , r) = \max \{ {u( x ) } : {| x- x _ {0} | = r } \} , $$

are convex functions in $ \mathop{\rm ln} r $ for $ n= 2 $, or in $ r ^ {2-} n $ for $ n \geq 3 $, on the interval $ r _ {1} \leq r \leq r _ {2} $; if $ u( x) $ is a subharmonic function in the disc (ball) $ 0 \leq r = | x- x _ {0} | \leq r _ {2} $, then, moreover, $ I( u; x _ {0} , r) $ and $ J( u; x _ {0} , r) $ are continuous non-decreasing functions in $ r $ on the interval $ 0 \leq r \leq r _ {2} $; moreover

$$ I( u; x _ {0} , 0) = J( u; x _ {0} , 0) = u( x _ {0} ); $$

in the latter instance,

$$ u( x _ {0} ) \leq J( u; x _ {0} , r) \leq I( u; x _ {0} , r) $$

for $ 0 \leq r \leq r _ {2} $. The mean values $ I( u; x _ {0} , r) $ and $ J( u; x _ {0} , r) $, considered as functions of the point $ x _ {0} $ for fixed $ u $ and $ r $, are subharmonic functions in the corresponding subdomain $ D ^ \prime \subset D $, and $ J( u; x _ {0} , r) $ is continuous. By forming iterations of a sufficiently-high order,

$$ u _ {m} ^ {(} k) ( x) = J \left ( u _ {m} ^ {(} k- 1) ; x, \frac{1}{m} \right ) ,\ \ u _ {m} ^ {(} 0) ( x) = u( x), $$

it is possible to obtain a monotone-decreasing sequence of subharmonic functions $ \{ u _ {m} ^ {(} k) ( x) \} _ {m= m _ {0} } ^ \infty $ of any degree of smoothness that converges, as $ m \rightarrow \infty $, to an arbitrarily given subharmonic function $ u( x) $.

The Newton potential and logarithmic potential of non-negative masses, when written with a minus sign, are subharmonic functions everywhere in the space $ \mathbf R ^ {n} $. On the other hand, one of the basic theorems in the theory of subharmonic functions is the Riesz local representation theorem: An arbitrary subharmonic function can be represented as the sum of a harmonic function and a potential with a minus sign (see ). More precisely, if $ u( x) $ is a subharmonic function in a domain $ D \subset \mathbf R ^ {n} $, then there exists a unique non-negative Borel measure $ \mu $ on $ D $( a measure associated with $ u( x) $, or a Riesz measure) such that for any compact set $ E \subset D $ the representation

$$ \tag{2 } u( x) = \int\limits _ { E } K( x- \xi ) d \mu ( \xi ) + h( x) $$

is valid, where $ K( x- \xi ) = \mathop{\rm ln} | x- \xi | $ when $ n= 2 $, $ K( x- \xi ) = - | x- \xi | ^ {2-} n $ when $ n \geq 3 $, and where $ h( x) $ is a harmonic function in the interior of $ E $. The Riesz theorem establishes a close link between the theory of subharmonic functions and potential theory.

If $ E $ is a regular closed domain $ \overline{G}\; $ bounded, for example, by a Lyapunov surface, and having a Green function $ g( x, \xi ) $, then as well as (2) a representation using the Green potential is valid:

$$ \tag{3 } u( x) = - \int\limits _ {\overline{G}\; } g( x, \xi ) d \mu ( \xi ) + w( x) , $$

where $ w( x) $ is the least harmonic majorant of the subharmonic function $ u( x) $ in the domain $ G $.

A representation in the form (3), generally speaking, does not hold in the whole domain of definition $ D $ of $ u( x) $, and in the theory of subharmonic functions great importance is attached to the question of distinguishing the class of subharmonic functions $ u( x) $ that allow a representation (3) in the whole domain $ D $, i.e. the question of distinguishing the class $ A $ of subharmonic functions $ u( x) $ that have a harmonic majorant in the whole domain $ D $. For example, if $ D = B( 0, R) $ is a ball (disc) and there exists a constant $ C = C( u) $ such that

$$ \tag{4 } \int\limits _ {S( 0,R) } u ^ {+} ( r \xi ) d \sigma ( \xi ) < C( u) < + \infty ,\ \ 0 < r < 1 , $$

then $ u( x) $ allows a representation (3) in $ D $, and the least harmonic majorant $ w( x) $ is, in turn, represented by a Poisson–Stieltjes integral:

$$ \tag{5 } w( x) = \int\limits \frac{R ^ {n-} 2 ( R ^ {2} - | x | ^ {2} ) }{| x- \xi | ^ {n} } d \nu ( \xi ), $$

where $ \nu $ is a Borel measure of arbitrary sign concentrated on the boundary sphere (circle) $ \partial D = S( 0, R) $( a boundary measure) and normalized by the condition $ \int d \nu ( \xi ) = 1 $.

With regard to (5), it is important in practical applications to know under which conditions the boundary measure $ \nu $ has only a negative singular component, i.e. under which conditions the component $ \nu ^ {+} $ in the canonical decomposition $ \nu = \nu ^ {+} - \nu ^ {-} $ is absolutely continuous. This question is answered by introducing the class of strictly-subharmonic functions (see [11][13], [15], as well as [10], where generalizations are also examined). Let $ \psi ( y) $ be an increasing concave function in $ y $ for which $ \lim\limits _ {y \rightarrow + \infty } y/ \psi ( y) = + \infty $. A function $ u( x) $, $ x \in D $, is said to be strictly subharmonic relative to $ \psi ( y) $ if $ \psi ( u( x)) $ is a subharmonic function. For example, logarithmically-subharmonic functions (cf. Logarithmically-subharmonic function) $ u( x) \geq 0 $, for which $ \mathop{\rm ln} u( x) $ is a subharmonic function, belong to the class of strictly-subharmonic functions. If condition (4) is fulfilled for a strictly-subharmonic function $ u( x) $ in the ball $ D $, then $ u( x) $ can be represented in the form (3) in $ D $, and the boundary measure $ \nu $ is characterized by the decomposition

$$ d \nu ( \xi ) = u( \xi ) d \sigma ( \xi ) - d \nu ^ {-} ( \xi ) ,\ \ \xi \in \partial D, $$

where $ u( \xi ) $ are the radial boundary values of the function $ u( x) $( which exist almost everywhere with respect to the Lebesgue measure on the sphere $ \partial D = S( 0, R) $), and $ \nu ^ {-} \geq 0 $ is the singular component of the measure $ \nu $.

Subharmonic functions of class $ A $ in the ball $ D $ have radial boundary values almost everywhere on $ \partial D = S( 0, R) $. However, examples have been constructed of bounded, continuous subharmonic functions in $ D $ that do not have non-tangential boundary values anywhere on $ \partial D $, a phenomenon that does not occur for harmonic functions. For non-tangential boundary values to exist, apart from (4) further conditions have to be imposed on the associated measure $ \mu $ in $ D $( see, for example, [14]).

One of the essential questions in the theory of subharmonic functions and its applications is the characterization of the boundary properties of functions of different subclasses of the class $ A $. The general method of introducing these subclasses consists of the fact that for strictly-subharmonic functions $ u( x) $ relative to a concave function $ \psi ( y) $, any non-decreasing function $ \phi ( y) $ is examined that is convex relative to $ \psi ( y) $ and that is such that $ \lim\limits _ {y \rightarrow + \infty } \phi ( y)/ \psi ( y) = + \infty $, and the class $ A _ \phi $ is introduced. For a sphere it is defined by the condition

$$ \int\limits _ {S( 0,R) } \phi ^ {+} ( u( r \xi )) d \sigma ( \xi ) < C( u, \phi ) < + \infty ,\ \ 0 < r < 1. $$

For the boundary properties of subharmonic functions, see [3][5], [10][16].

For functions that can be represented as the difference between two subharmonic functions, the concept of characteristic in the sense of R. Nevanlinna has been introduced, and the theory of functions of bounded characteristic (cf. Function of bounded characteristic) has been generalized (see [3], ).

A distinctive generalization of the theory of entire functions (cf. Entire function) is the theory of subharmonic functions in the whole space $ \mathbf R ^ {n} $. Here, generalizations of the Weierstrass and Hadamard classical representation theorems of entire functions have been obtained, along with the theory of the growth and value distribution, the theory of asymptotic values and asymptotic paths, etc. (see ).

In the theory of analytic functions of several complex variables, the study of the subclasses of plurisubharmonic functions and pluriharmonic functions (cf. Plurisubharmonic function; Pluriharmonic function) is of considerable importance (see [17]). For axiomatic generalizations of subharmonic functions, see [9].

References

[1] F. Hartogs, "Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen welche nach Potenzen einer Veränderlichen fortschreiten" Math. Ann. , 62 (1906) pp. 1–88
[2a] F. Riesz, "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel I" Acta Math. , 48 (1926) pp. 329–343
[2b] F. Riesz, "Sur les fonctions sousharmoniques et leur rapport à la théorie du potentiel II" Acta Math. , 54 (1930) pp. 321–360
[3] I.I. Privalov, "Subharmonic functions" , Moscow-Leningrad (1937) (In Russian)
[4] I.I. Privalov, "Boundary properties of single-valued analytic functions" , Moscow (1941) (In Russian)
[5] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[6] T. Radó, "Subharmonic functions" , Springer (1937)
[7a] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , 1 , Acad. Press (1976)
[7b] W.K. Hayman, "Subharmonic functions" , 2 , Acad. Press (1989)
[8] M. Brelot, "Etude des fonctions sousharmoniques au voisinage d'un point" , Hermann (1934)
[9] M. Brélot, "Lectures on potential theory" , Tata Inst. (1967)
[10] M. Heins, "Hardy classes on Riemann surfaces" , Springer (1969)
[11] E.D. Solomentsev, "On some classes of subharmonic functions" Izv. Akad. Nauk SSSR , 5–6 (1938) pp. 571–582 (In Russian)
[12] I.I. Privalov, P.I. Kuznetsov, "Sur les problèmes limites et les classes différentes de fonctions harmoniques et sousharmoniques définies dans une domaine arbitraire" Mat. Sb. , 6 : 3 (1939) pp. 345–376 (In Russian) (French abstract)
[13] E.D. Solomentsev, "Classes of functions subharmonic on a half-space" Vestnik Moskov. Gos. Univ. Ser. Mat.-Mekh. Astron. : 5 (1959) pp. 73–91 (In Russian)
[14] E.D. Solomentsev, "On boundary values of subharmonic functions" Czech. Math. J. , 8 : 4 (1958) pp. 520–536 (In Russian) (French abstract)
[15] L. Gårding, L. Hormander, "Strongly subharmonic functions" Math. Scand. , 15 (1964) pp. 93–96
[16] E.D. Solomentsev, "Harmonic and subharmonic functions and their generalization" Itogi Nauk Mat. Anal. Teor. Veroyatnost. Regulirovanie 1962 (1964) pp. 83–100 (In Russian)
[17] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

Axiomatic potential theory can be founded upon the properties 1) and 3), completed by some additional properties of the set $ S $ of negative subharmonic functions: a) the upper semi-continuous regularization of the supremum of a subset of $ S $ is subharmonic; b) (the Riesz decomposition property) for any $ u , v, w \in S $, $ u \geq v+ w $, there exist $ v ^ \prime , w ^ \prime \in S $ such that $ u= v ^ \prime + w ^ \prime $, $ v ^ \prime \geq v $, $ w ^ \prime \geq w $; and c) any $ u \in S $ is the limit of a decreasing sequence in a sufficiently nice subcone of continuous subharmonic functions. See [a1].

References

[a1] J. Bliedtner, W. Hansen, "Potential theory. An analytic and probabilistic approach to balayage" , Springer (1986)
[a2] L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)
[a3] O.D. Kellogg, "Foundations of potential theory" , Dover, reprint (1954) pp. 315ff (Re-issue: Springer, 1967)
How to Cite This Entry:
Subharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subharmonic_function&oldid=48893
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article