Circle log-mean of a holomorphic function ( $L_{f} (R) = \frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ$ )

Under the hypotheses of Jensen's formula, the circle log-mean is the angular average of the natural logarithm of the complex modulus. The factor

1/ (2 π)

normalizes the integral to a mean. It averages values at equally weighted angles on the circle, and Jensen's formula computes it from

∣ f (0) ∣

and the interior zeros of a function.

Table of contents

Circle geometric mean of a holomorphic function ( $G_{f} (R) = exp L_{f} (R)$ )

 0  0

The continuous angular geometric mean of the positive boundary complex modulus is

G_{f} (R) = exp (\frac{1}{2 π} \int_{0}^{2 π} lo g ∣ f (R e^{i θ}) ∣ d θ) = ∣ f (0) ∣ \prod_{j = 1}^{N} \frac{R}{∣ a _{j} ∣} .

(1)

Here the empty product is one. This differs from the angular arithmetic mean

A_{f} (R) = (2 π)^{- 1} \int_{0}^{2 π} ∣ f (R e^{i θ}) ∣ d θ

. In general

G_{f} (R) \leq A_{f} (R)

by the concavity of the logarithm; equality requires constant boundary complex modulus. For a holomorphic function with no zeros of a function in the closed disk, Jensen's formula gives

G_{f} (R) = ∣ f (0) ∣

, even when the boundary complex modulus varies with angle. Thus replacing the mean of a logarithm by the logarithm of an arithmetic mean loses the exact zero identity.

Polynomial circle log-mean ( $L_{p} (R) = lo g ∣ c ∣ + \sum_{j} lo g max (R, ∣ a_{j} ∣)$ )

 0  0

For a polynomial

p (z) = c \prod_{j} (z - a_{j})

with

c \neq = 0

, no polynomial root at zero, and

R > 0

different from every

∣ a_{j} ∣

, Jensen's formula gives the displayed expression. The polynomial roots are repeated according to their multiplicity of a zero: an exterior root contributes

lo g ∣ a_{j} ∣

, whereas an interior root contributes

lo g R

For an explicit example, take the polynomial

f (z) = 3 (z - \frac{1}{4}) (z + \frac{1}{2}), f (0) = - \frac{3}{8} .

(1)

Its two simple zeros are

1/4

and

- 1/2

. The circle log-mean of a holomorphic function is therefore

L_{f} (R) = ⎩ ⎨ ⎧ lo g (3/8), lo g (3 R /2), lo g (3 R^{2}), 0 < R < 1/4, 1/4 < R < 1/2, R > 1/2.

(2)

R = 0.2

the open disc contains no zeros of a function, at

R = 0.4

it contains

1/4

only, and at

R = 0.75

it contains both. These three circles avoid the polynomial roots; the radii

1/4

and

1/2

are excluded from the displayed version.

Figure 1.
The zeros of $3 (z - 1/4) (z + 1/2)$ inside three concentric disks
. The three boundary circles enclose respectively zero, one and two simple zeros. Each zero contributes to Jensen's formula only after the radius exceeds its complex modulus. **Details**.

The corresponding values of the circle log-mean of a holomorphic function and the circle geometric mean of a holomorphic function are

R 0.2 0.4 0.75 n_{f} (R) 012 L_{f} (R) lo g (3/8) \approx - 0.980829 lo g (3/5) \approx - 0.510826 lo g (27/16) \approx 0.523248 G_{f} (R) 3/8 3/5 27/16

(3)

For example, at

R = 0.4

the contribution of the sole interior zero of a function is

lo g (0.4/0.25) = lo g (8/5)

; adding it to

lo g (3/8)

gives

lo g (3/5)

. The second polynomial root still affects

f (0)

, but contributes no additional interior-zero term at this radius.

Figure 2.
Circle log-mean versus log-radius for $3 (z - 1/4) (z + 1/2)$
. The log-radius slope of the circle log-mean is zero, one and two in the three intervals, matching the radial zero-counting function. Open markers exclude the boundary-zero radii $1/4$ and $1/2$ ; filled markers show the three computed examples. **Details**.

Log-radius slope of the circle log-mean ( $\frac{d}{d l o g R} L_{f} (R) = n_{f} (R)$ )

 0  0

Fix a holomorphic function on a neighbourhood of a closed disk, with

f (0) \neq = 0

. On an interval of positive radii whose circles contain no zeros of a function, differentiating Jensen's formula with respect to

x = lo g R

gives

\frac{d}{d x} L_{f} (e^{x}) = n_{f} (e^{x}) .

(1)

Each interior zero of a function contributes slope one for each unit of its multiplicity of a zero. Between consecutive zero moduli,

L_{f} (e^{x})

is an affine function; passing the modulus of a zero of multiplicity of a zero

m

increases the adjacent slopes by

m

. The finite sum

lo g ∣ f (0) ∣ + \sum_{j} max (0, x - lo g ∣ a_{j} ∣)

defines a continuous convex function of

x

whose values agree with the circle log-mean of a holomorphic function at every boundary-zero-free radius. This describes the adjacent line segments without asserting a boundary-singularity extension of the integral.