subharmonic and superharmonic functions

subharmonic and superharmonic functions

First let’s look at the most general definition.

Definition

.

Let

G

⊂

ℝ

n

and let

φ

:

G

→

ℝ

∪

{

-

∞

}

be an

upper semi-continuous

function

,
then

φ

is

subharmonic

if for every

x

∈

G

and

r

>

0

such that

B

⁢

(

x

,

r

)

¯

⊂

G

(the

closure

of the open ball of radius

r

around

x

is still in

G

) and every real valued

continuous function

h

on

B

⁢

(

x

,

r

)

¯

that is

harmonic

in

B

⁢

(

x

,

r

)

and

satisfies

φ

⁢

(

x

)

≤

h

⁢

(

x

)

for all

x

∈

∂

⁡

B

⁢

(

x

,

r

)

(boundary of

B

⁢

(

x

,

r

)

) we have that

φ

⁢

(

x

)

≤

h

⁢

(

x

)

holds for all

x

∈

B

⁢

(

x

,

r

)

.

Note that by the above, the function which is identically

-

∞

is subharmonic, but some authors exclude this function by definition.
We can define

superharmonic

functions in a

similar

fashion to get that

φ

is superharmonic if and only if

-

φ

is subharmonic.

If we restrict our domain to the

complex plane

we can get the following definition.

Definition

.

Let

G

⊂

ℂ

be a region and let

φ

:

G

→

ℝ

be a continuous function.

φ

is said to
be

subharmonic

if whenever

D

⁢

(

z

,

r

)

⊂

G

(where

D

⁢

(

z

,

r

)

is a closed
disc around

z

of radius

r

) we have

φ

⁢

(

z

)

≤

1

2

⁢

π

⁢

∫

0

2

⁢

π

φ

⁢

(

z

+

r

⁢

e

i

⁢

θ

)

⁢

𝑑

θ

,

and

φ

is said to be

superharmonic

if whenever

D

⁢

(

z

,

r

)

⊂

G

we have

φ

⁢

(

z

)

≥

1

2

⁢

π

⁢

∫

0

2

⁢

π

φ

⁢

(

z

+

r

⁢

e

i

⁢

θ

)

⁢

𝑑

θ

.

Intuitively what this means is that a subharmonic function is at any point
no

greater

than the

average

of the values in a circle around that point. This implies that a non-constant subharmonic function does not achieve its

maximum

in a region

G

(it would achieve it at the boundary if it is

continuous

there). Similarly for a superharmonic
function, but then a non-constant superharmonic function does not achieve its
minumum in

G

. It is also easy to see that

φ

is subharmonic if and only if

-

φ

is superharmonic.

Note that when equality always holds in the above equation then

φ

would
in fact be a

harmonic function

. That is, when

φ

is both subharmonic and
superharmonic, then

φ

is harmonic.

It is possible to relax the

continuity

statement above to take

φ

only upper semi-continuous in the subharmonic case and lower semi-continuous in the
superharmonic case. The

integral

will then however need to be the

Lebesgue
integral

(

http://planetmath.org/Integral2

) rather than the

Riemann integral

which may not be defined for such
a function. Another thing to note here is that we may take

ℝ

2

instead of

ℂ

since we never did use

complex multiplication

. In that case however we must rewrite the expression

z

+

r

⁢

e

i

⁢

θ

in of the
real and

imaginary parts

to get an expression in

ℝ

2

.

It is also possible generalize the range of the functions as well. A subharmonic function could have a range of

ℝ

∪

{

-

∞

}

and a superharmonic function could have a range of

ℝ

∪

{

∞

}

. With this

generalization

, if

f

is a

holomorphic function

then

φ

⁢

(

z

)

:=

log

⁡

|

f

⁢

(

z

)

|

is a subharmonic function if we
define the value of

φ

⁢

(

z

)

at the zeros of

f

as

-

∞

.
Again it is important to note that with this
generalization we again must use the Lebesgue integral.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

2

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

subharmonic and superharmonic functions

Canonical name

SubharmonicAndSuperharmonicFunctions

Date of creation

2013-03-22 14:19:39

Last modified on

2013-03-22 14:19:39

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

12

Author

jirka (4157)

Entry type

Definition

Classification

msc 31C05

Classification

msc 31A05

Classification

msc 31B05

Related topic

HarmonicFunction

Defines

subharmonic

Defines

subharmonic function

Defines

superharmonic

Defines

superharmonic function

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