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$∣ z ∣^{2}$ and $\log ∣ z ∣$ are the model basic subharmonic functions

Example

Two standard examples on the plane are:

on all of $C$ , the function $u (z) = ∣ z ∣^{2} = x^{2} + y^{2}$ ;
on $C$ , the function $v (z) = \log ∣ z ∣$ with the convention $v (0) = - \infty$ .

Both are subharmonic, and $v$ is harmonic away from $0$ .

Facts & Assumptions

Given: The functions $u (z) = ∣ z ∣^{2}$ on $C$ and $v (z) = \log ∣ z ∣$ with $v (0) = - \infty$ .

[L1]

A $C^{2}$ real function is subharmonic exactly when its Laplacian is nonnegative (A C^2 function is subharmonic exactly when its Laplacian is nonnegative).

[L2]

For a holomorphic function, the logarithm of the modulus is subharmonic (The logarithm of the modulus of a holomorphic function is subharmonic).

Verification

technique · direct

1.1L1givenalgebra

Writing $z = x + i y$ , one has $u (x + i y) = x^{2} + y^{2}$ , so [L1, given, algebra] $Δ u = u_{x x} + u_{y y} = 2 + 2 = 4 \geq 0.$ By [L1], $u$ is subharmonic on $C$ .

2.1L2given∎

The identity map $f (z) = z$ is holomorphic on $C$ , and [L2, given] $\log ∣ f (z) ∣ = \log ∣ z ∣$ with value $- \infty$ at the zero $z = 0$ . Therefore [L2] makes $v$ subharmonic on $C$ . Away from $0$ , the function $v$ is the real part of the holomorphic logarithm and so is harmonic there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Harold P. Boas, Class Notes Math 618: Complex Variables II, Spring 2016 (standard reference, not scraped)

∣z∣2∣z∣2 and log⁡∣z∣log∣z∣ are the model basic subharmonic functions