Radial zero-counting function ( $n_{f} (r) = \sum_{∣ a ∣ < r} ord_{a} f$ )

For a nonzero holomorphic function,

n_{f} (r)

counts its zeros of a function in the centred open disc of radius

r

, with multiplicity of a zero. Thus a double zero counts twice, even though it is a single point. On every smaller closed disk in the domain the count is finite. The strict inequality

∣ a ∣ < r

specifies the convention at a zero modulus; elsewhere it agrees with counting in the closed disk. In the polynomial circle log-mean example the count is zero, one and two on the three displayed intervals.

Table of contents
- Integrated zero count Radial zero-counting function

Integrated zero count ( $N_{f} (R) = \int_{0}^{R} \frac{n _{f} ( t )}{t} d t$ )

 0  0

Under the hypotheses of Jensen's formula,

f (0) \neq = 0

ensures that the radial zero-counting function vanishes for sufficiently small positive

t

. Integrating each zero's contribution separately gives

N_{f} (R) = \int_{0}^{R} \frac{n _{f} ( t )}{t} d t = \sum_{∣ a ∣ < R} ord_{a} f \int_{∣ a ∣}^{R} \frac{d t}{t} = \sum_{∣ a ∣ < R} ord_{a} f lo g \frac{R}{∣ a ∣} = L_{f} (R) - lo g ∣ f (0) ∣.

(1)

The values of

n_{f}

at its finitely many jump points do not change the integral. For

0 < r < R

, every zero in the smaller open disc supplies at least

lo g (R / r)

per unit of multiplicity of a zero, so

n_{f} (r) lo g (R / r) \leq N_{f} (R) \leq lo g M_{f} (R) - lo g ∣ f (0) ∣, M_{f} (R) = max_{∣ z ∣ = R} ∣ f (z) ∣.

(2)

The second inequality bounds the circle log-mean of a holomorphic function by the logarithm of the largest boundary complex modulus, recovering the Jensen zero-count bound. An upper bound on boundary growth and a fixed nonzero centre value therefore constrain interior zeros. The factor