Definition:Circumferential Mean

Definition

Let $D$ be a complex domain.

Let $f: D \to \R$ be a subharmonic function or a superharmonic function.

Let $\map {B^-} {a; r} \subseteq D$ be a closed disk with center $a$ and radius $r$:

The integral over the boundary of $\map {B^-} {a; r}$:

$\ds \int_0^{2 \pi} \map f {a + r e^{i \theta} } \rd \theta$

is known as the circumferential mean of $f$ over $\map {B^-} {a; r}$.

Also see

• Results about circumferential means can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): circumferential mean
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): subharmonic function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circumferential mean
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): subharmonic function